SDP Based Approach for Graph

Partitioning and

Embedding Negative Type Metrics into

L1

Subhash Khot (Georgia Tech)

Nisheeth K Vishnoi (IBM Research and Georgia Tech)

CS Perspective

Math Perspective

Parts I amp II

CS Story SparsityS

Sc

Sparsity of a Cut|E(SSc)|----------|S| |Sc|

Sparsest Cut (SC)cut of minimum sparsity b-Balanced Separator (BS)

cut with |S||Sc| ge bn that minimizes |E(SSc)|

b ~ frac12

ApplicationsRelated Measures

Sparsity is often referred to as Graph Conductance

Edge expansion or isoperimetric constant

Applications VLSI Layout

Clustering

Markov Chains

Geometric (Metric) Embeddings

Estimating Sparsity

λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆

λ2 spectral gap (second eigenvalue of the Laplacian)

Not satisfactory eg n-cycle

Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G

a(G) le sparsity(G) le a(G) f(n)

f(n)=1 is hardWhat about f(n) = log n or even 10

Hard to compute exactly ndash compute ldquoapproximationsrdquo

HistoryAlgorithms

Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)

O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)

O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)

O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)

Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)

Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

CS Story SparsityS

Sc

Sparsity of a Cut|E(SSc)|----------|S| |Sc|

Sparsest Cut (SC)cut of minimum sparsity b-Balanced Separator (BS)

cut with |S||Sc| ge bn that minimizes |E(SSc)|

b ~ frac12

ApplicationsRelated Measures

Sparsity is often referred to as Graph Conductance

Edge expansion or isoperimetric constant

Applications VLSI Layout

Clustering

Markov Chains

Geometric (Metric) Embeddings

Estimating Sparsity

λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆

λ2 spectral gap (second eigenvalue of the Laplacian)

Not satisfactory eg n-cycle

Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G

a(G) le sparsity(G) le a(G) f(n)

f(n)=1 is hardWhat about f(n) = log n or even 10

Hard to compute exactly ndash compute ldquoapproximationsrdquo

HistoryAlgorithms

Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)

O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)

O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)

O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)

Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)

Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

ApplicationsRelated Measures

Sparsity is often referred to as Graph Conductance

Edge expansion or isoperimetric constant

Applications VLSI Layout

Clustering

Markov Chains

Geometric (Metric) Embeddings

Estimating Sparsity

λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆

λ2 spectral gap (second eigenvalue of the Laplacian)

Not satisfactory eg n-cycle

Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G

a(G) le sparsity(G) le a(G) f(n)

f(n)=1 is hardWhat about f(n) = log n or even 10

Hard to compute exactly ndash compute ldquoapproximationsrdquo

HistoryAlgorithms

Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)

O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)

O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)

O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)

Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)

Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Estimating Sparsity

λ2(G)n le sparsity(G) le 3n radicλ2(G) radic∆

λ2 spectral gap (second eigenvalue of the Laplacian)

Not satisfactory eg n-cycle

Approx Algo For any graph G on n vertices compute a(G) which is within a mult factor f(n)ge1 of sparsity of G

a(G) le sparsity(G) le a(G) f(n)

f(n)=1 is hardWhat about f(n) = log n or even 10

Hard to compute exactly ndash compute ldquoapproximationsrdquo

HistoryAlgorithms

Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)

O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)

O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)

O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)

Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)

Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

HistoryAlgorithms

Spectral Graph partitioning Alon-Milman rsquo85 Speilman-Teng rsquo96(eigenvector based)

O(log n) Leighton-Rao rsquo88 (Linear Programming (LP) based)

O(log n) London-Linial-Rabinovitch rsquo94 Aumann-Rabani rsquo94 (connection to metric embeddings)

O(radiclog n) Arora-Rao-Vazirani lsquo04(Semi-Definite Programming (SDP) based)

Hardness NP-Hard Hard to approximate within any constant factor (assuming UGC)

Chawla-Krauthgamer-Kumar-Rabani-Sivakumar lsquo05 Khot-Vishnoi lsquo05

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Quadratic Program for BS

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic ProgramInput G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

SDP for Balanced Separator

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Why is this a Relaxation SDP Relaxation

G(VE)

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Relaxation

u be a unit vector and (SSc)|S| |Sc| = n2

For i є S vi = u i є Sc vi = - u ||vi-vj||2 = 4 δS(ij) Cost of solution = |E(SSc)| sdp le opt

SDP can be computed in polynomial time

Boils down to the spectral approach Nothing gained()

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Quadratic Program for BS hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2

+ |vj - vk |2 |vi - vk|2 (redundant)

Quadratic Program

Input G(VE)

Output (SSc)

st

|S||Sc| = n2

which minimizes

|E(SSc)|

Balanced Separator

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

SDP for Balanced Separator hellip

i vi -11

Minimize frac14 |vi - vj |2

ij ε E

|vi - vj |2 = n2

iltj

Triangle Inequality i j k |vi - vj |2+|vj - vk |2|vi - vk|2

Quadratic Program SDP Relaxation

i vi Rn ||vi||=1

Minimize frac14 || vi - vj ||2

ij E

Well-Separatedness

|| vi - vj ||2 = n2

iltj

Triangle Inequality i j k ||vi - vj||2 +||vj - vk||2 ||vi - vk||2

Still a relaxation hellip

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Geometry of Triangle Ineq

le 90ovi

vk

vj

each step of length 1

t-steps length at-most radict

Rules out the embedding obtained by the spectral method

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Integrality Gap Upper Bound

Arora-Rao-Vazirani rsquo04

O(radiclog n) for Sparsest Cut Balanced Separator

sdp within a factor of O(radiclog n) of the opt

Integrality gap max over all graphs on n vertices the ratio of

optsdp (as a function of n)

ARV conjectured that the integrality gap is upper bounded by some constant (independent of n)

Lack of any counterexample

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection

Part II 1 Integrality Gap instance

Hypercube and Cutsbull Kahn-Kalai-Linial (isoperimetry of hypercube)

The GraphThe SDP Solution

2 Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Math Story Metric Embeddings

bull Metric is a distance function d on [n] x [n] st d(i j) + d(j k) d(i k)

(triangle inequality)

bull Metric d embeds into metric with distortion 1 if there is a map φ st

ij d(i j) (φ(i) φ(j)) d(i j) (distances are preserved upto a factor of )

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Negative Type Metrics (squared-L2)

bull d on 12hellipn is of negative type if there are vectors v1 v2 hellip vn

bull Such that d(i j) = || vi - vj ||2 satisfies triangle inequality

bull Same as i j k || vi - vj ||2

+ || vj - vk ||2 || vi - v k||2

bull NEG = class of such metrics bull arise as SDP solutions

bull L1 NEG

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Embedding NEG into L1

Conjecture (Explicit by Goemans Linial abt rsquo95)

Every NEG metric embeds into L1 with

O(1) (constant) distortion

Whatrsquos the connection to sparsity

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Cuts and L1 Metrics

pS non-negative real for every subset S of [n]

d(ij) = pSδS(ij)

Fact d is isometrically embeddable in L1

Further Every L1 embeddable metric on n-points can be written as non-negative linear sum of cut-metrics on 1hellipn

Cut-Metrics on 12hellipn

δS(i j) = 1 if i j are separated by (SSc) = 0 otherwise

S Sc

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Sparsest Cut asymp Optimizing Over L1

[Aumann Rabani 98 Linial London Rabinovich rsquo94]

bull LP-relaxation over all METRICSbull [Bourgain rsquo85] Every n-point metric embeds

into L1 with O(log n) distortionbull O(log n) factor approximation for Sparsity

i~j δS(i j)

---------- iltj δS(i j)

Minimize S V

Minimize d is L1

i~j d(i j)

---------- iiltj d(i j)

=

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Metric Embeddings amp Sparsity

bull Optimizing over cuts asymp Optimizing over L1 metrics

bull SDP solution asymp Optimizing over NEG

bull Goemans-LinialARV Conjecture NEG embeds in L1 with O(1) distortion Integrality Gap is O(1)

bull Implies O(1) approx algo for estimating sparsity

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani lsquo06Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Outline of this Talk Part I

1 Graph PartitioningMotivation amp HistorySDP Approach

2 Embedding Negative Type Metrics into L1 Metric Spaces and EmbeddabilityCut Cone asymp L1

Negative Type Metrics as SDP SolutionsLLRAR Connection Part II

1 Integrality Gap instance Hypercube and Cuts

bull Kahn-Kalai-Linial (isoperimetry of hypercube)The GraphThe SDP Solution

2 Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Recall Integrality Gap Lower Bound

Sparsest Cut Balanced Separator

log log n integrality gap instance

Khot-Vishnoi rsquo05 Krauthgamer-Rabani rsquo06 Devanur-Khot-Saket-Vishnoi lsquo06

bull Disproves the GLARV Conjecture bull Previous best lower bound 116 [Zatloukal rsquo04]

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Starting Point Hypercube()

H=-11k

n = 2k

(111)

(-111)

(11-1)

(1-11)

(1-1-1)

(-1-11)

(-11-1) (-1-1-1)

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Hypercube hellip H=-11k Advantages

bull Understand cuts in H tools from Fourier Analysis

bull Vertex is a vector in Rk starting point for SDP solution

But hellip

bull Hypercube has ldquosmallrdquo balanced cuts coordinate cuts have 1k fraction of the edges

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Cuts in Hypercube Coordinate of edges = |E(H)|k

Edges across pairs of vertices differing in i-th bit

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Cuts in Hypercube hellip

decompose into coordinate cuts

any balanced cut has a coord cut which contributes E(H)k2 edges

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Kahn-Kalai-Linial

any balanced cut has a coordinate cut which contributes E(H) (log

k)k2 edges

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Increasing Size of Balanced Cuts

consider balanced cuts in which coordinates are indistinguishable

(wrt to their contribution to the cut)

can be achieved by symmetrizing the hypercube

each coordinate contributes equally total E(H) (log k)k edges

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

eg 4-dim hypercube

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

More Formally hellipH=-11k with a rotation group acting on its coordinates

Partitions H into equivalence classes V1hellipVn

Each Vi is a vertex Edges are hypercube edges

G(VE) |E(G)|=|E(H)| k ~ log n

Balanced cuts in G correspond to balanced cuts in H

KKL any balanced cut (in H) has ldquoardquo coordinate cut which contributes E(H) (log k)k2 edges to the cut

Group is transitive ldquoeveryrdquo coordinate cut has the same contribution

Any balanced cut in G has ge E(G) (log k)k = E(G) (log log n)(log n)

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Integrality Gap Lower Bound

Thm Construct graph G(1hellipnE) and unit vector assignment i -gt vi є Rn st

1 G is an ldquoexpanderrdquo every frac14 balanced cut has Ω(|E|(log log n)log n) edges (Kahn-Kalai-Linial)

2 ldquoLowrdquo SDP solution O(|E|log n)

3 Well-Separatedness Σiltj ||vi-vj||2 = n2

4 Triangle inequality d(ij)=||vi-vj||2 is a metric

Integrality gap Ω(log log n)

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

SDP Solution

(1111)

(-1-1-1-1)

(111-1) (-1111) (1-111) (11-11)

(-1-1-11) (1-1-1-1) (-11-1-1) (-1-11-1)

(11-1-1) (-111-1) (-1-111) (1-1-11)

(1-11-1) (-11-11)

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Formally SDP SolutionVertex equivalence class x1 x2 hellip xk (rotations)

Vector (1radick) Σj xj

Observations

bull Edge across two nodes differing in one bit

Contribution to sdp ~ 1k sdp le |E(G)|k = |E(G)|(log n)

Triangle Inequality (little bit of work and case analysis)

For most classes x1 hellip xk is ldquonearly orthogonalrdquo

Hidden Gram-Schmidt Orthogonalization Tensoring Well-separatedness

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion

Conclusion

bull (Simple) log log n integrality gap for SCBS

bull Close the gap between log log n and log n

bull [Lee-Naor rsquo06 Cheeger-Kleiner rsquo06] Another counter-example May give (log n)c

Thank you

• Slide 1
• CS Story Sparsity
• Applications
• Estimating Sparsity
• History
• Outline of this Talk
• Slide 7
• Quadratic Program for BS
• SDP for Balanced Separator
• Why is this a Relaxation
• Quadratic Program for BS hellip
• SDP for Balanced Separator hellip
• Geometry of Triangle Ineq
• Integrality Gap Upper Bound
• Slide 15
• Math Story Metric Embeddings
• Negative Type Metrics (squared-L2)
• Embedding NEG into L1
• Cuts and L1 Metrics
• Sparsest Cut asymp Optimizing Over L1
• Metric Embeddings amp Sparsity
• Integrality Gap Lower Bound
• Slide 23
• Slide 24
• Recall Integrality Gap Lower Bound
• Slide 26
• Starting Point Hypercube()
• Hypercube hellip
• Cuts in Hypercube Coordinate
• Cuts in Hypercube hellip
• Kahn-Kalai-Linial
• Increasing Size of Balanced Cuts
• eg 4-dim hypercube
• More Formally hellip
• Slide 35
• SDP Solution
• Formally SDP Solution
• Conclusion