Spectrally Thin Trees

Spectrally Thin Trees

Nick Harvey University of British Columbia

Joint work with Neil Olver (MIT Vrije Universiteit)

Approximating Dense Objectsby Sparse Objects

Floor joists

Wood Joists Engineered Joists


Bridges

Masonry Arch Truss Arch


Bones

Human Femur Robin Bone


Graphs

Dense Graph Sparse Graph

How well can any graph be approximated by a sparse graph?

First way to compare graphsDo graphs have nearly same weight on

corresponding cuts?

S S

Second way to compare graphsDo their Laplacian matrices have nearly same

eigensystem?

5 -1 -1 -1 -1 -14 -1 -1 -1 -1

-1 -1 6 -1 -1 -1 -1-1 5 -1 -1 -1 -1

-1 -1 -1 7 -1 -1 -1 -1-1 -1 -1 5 -1 -1-1 -1 -1 5 -1 -1

-1 -1 -1 -1 6 -1 -1-1 -1 -1 -1 -1 5

-1 -1 -1 -1 -1 -1 6

6 -1 -55 -1 -3 -1

-1 2 -18 -8

-1 2 -11 -1

-3 -1 5 -12 -1 -1

-5 -1 -1 -1 8-1 -8 -1 10

First way, more formally

Weight of cut: u(±(S)) w(±(S))

S

Edge weights u

S

Edge weights w

®-cut sparsifier: u(±(S)) · w(±(S)) · ®¢u(±(S)) 8S

Cut ±(S) = { edge st : s2S, tS }

Second way, more formally

Lu = D-A =

7 -2 -5-2 3 -1-5 -1 16 -

10-

1010

abcd

a b c d

weighted degree of node

c

negative of u(ac)

Graph with weights u:ab

dc5 102 1

Laplacian Matrix:

Second way, more formally

Def: A¹B , B-A is PSD , xTAx · xTBx 8x2Rn

®-spectral sparsifier: Lu ¹ Lw ¹ ®¢Lu

5 -1 -1 -1 -1 -14 -1 -1 -1 -1

-1 -1 6 -1 -1 -1 -1-1 5 -1 -1 -1 -1

-1 -1 -1 7 -1 -1 -1 -1-1 -1 -1 5 -1 -1-1 -1 -1 5 -1 -1

-1 -1 -1 -1 6 -1 -1-1 -1 -1 -1 -1 5

-1 -1 -1 -1 -1 -1 6

6 -1 -55 -1 -3 -1

-1 2 -18 -8

-1 2 -11 -1

-3 -1 5 -12 -1 -1

-5 -1 -1 -1 8-1 -8 -1 10

Edge weights u

Edge weights w

Lu = Lw =

Thin trees

Let w be supported on a spanning tree®-thin tree: w(±(S)) · ®¢u(±(S)) 8S®-spectrally thin tree: Lw ¹ ®¢Lu

S

Edge weights u

S

Edge weights w

Connectivity and Conductance

Connectivity: kst = min { u(±(S)) : s2S, tS }Global connectivity: K = min { ke : e2E }Effective Resistance from s to t: voltage difference when a 1-amp current source placed between s and tEffective Conductance: cst = 1 / (effective resistance from s to t)Global conductance: C = min { ce : e2E }Fact: cst · kst 8s,t.Example: cst =1/n but kst=1.Long paths affect conductance but not connectivity

Various Kappas: κ κ κ κ κ κ κ κ κ κκκ κ κ

s t

Motivation for thin trees

Goddyn’s Conjecture: every graph has a O(1/K)-thin tree

O(1)-approximation for asymmetric TSPJaeger’s conjecture on nowhere-zero 3-flows [solved]Goddyn-Seymour conjecture on nowhere-zero 2+² flows

Spectrally thin trees may be a useful step towards thin trees

Edge weights u

Unweighted

Intriguing Phenomenon

cut-sparsifier result involving connectivities holds

seemingly if and only if

spectral-sparsifier result involving conductances holds

Uniform sampling

Recall K = min { ke : e2E }Karger Skeletons:

Define p = O( ²-2 log(n) / K )Sample every edge e with probability pGive every sampled edge e weight 1/p

Resulting graph is a (1+²)-cut sparsifier,and number of edges shrinks by factor O(p), whp.Spectral version: [unpublished]Replace K by C and “cut” by “spectral”

and C = min { ce : e2E }

spectral

C

Assume unweighted

Uniformsampling

Cutsparsifier,

connectivityweights

Spectralsparsifier,conducta

nceweights

Karger

Unpublished

Non-uniform sampling

Let ke be “strong connectivity” of edge eBenczur-Karger:

Define pe = O( ²-2 log(n) / ke )Sample every edge e with probability pe

Give every sampled edge e weight 1/pe

Resulting graph is a (1+²)-cut sparsifier andnumber of sampled edges is O(n log(n) ²-2), whp.Fung-Hariharan-Harvey-Panigrahi:Replace ke by ke and log(n) by log2(n).

ke

log2(n)

log2(n)*

*

*

Open QuestionImprove to

log(n)

Non-uniform sampling

Let ke be “strong connectivity” of edge eBenczur-Karger:

Define pe = O( ²-2 log(n) / ke )Sample every edge e with probability pe

Give every sampled edge e weight 1/pe

Resulting graph is a (1+²)-cut sparsifier andnumber of sampled edges is O(n log(n) ²-2), whp.Spielman-Srivastava:Replace ke by ce and “cut” by “spectral”.

ce

spectral sparsifier

*

*

*

Uniformsampling

Cutsparsifier,

connectivityweights


nceweights

Karger

Unpublished

Non-uniformsamplingBenczur-Karger

Spielman-Srivastava

Fung- Hariharan-

Harvey-Panigrahi

Thin trees

Asadpour et al:Pick special distribution on spanning trees such thatevery edge e has Pr[ e in tree ] = £( 1/ K )Give every edge e in tree weight K

Resulting tree is an -cut thin treeMaximum entropy distribution worksChekuri et al: Pipage rounding also worksHarvey-Olver: Replace K by ce and “cut” by “spectral”

cecespectrally thin

Uniformsampling

Cutsparsifier,

connectivityweights


nceweights

Karger

Unpublished


Spielman-Srivastava

Fung- Hariharan-

Harvey-Panigrahi

O(log n / log log n)thin trees

Asadpouret al.

Harvey-Olver

Chekuri-Vondrak-

Zenklusen

Linear-size sparsifiers

Batson-Spielman-Srivastava:Can efficiently construct a (1+²)-spectral sparsifierwith O( n²-2

) edges such that “on average”weight of each edge e is £( ²2

ce )Marcus-Spielman-Srivastava: Remove “on average”, but not efficient.Open question:Replace ce by ke and “spectral” by “cut”?

ke?

cut?

Uniformsampling

Cutsparsifier,

connectivityweights


nceweights

Karger

Unpublished


Spielman-Srivastava

Fung- Hariharan-

Harvey-Panigrahi


Asadpouret al.

Harvey-Olver

Linear-sizeSparsifiers

Batson-Spielman-Srivastava

Marcus-Spielman-Srivastava

?Chekuri-Vondrak-

Zenklusen

Optimal thin trees

Suppose we have a (1+²)-spectral sparsifier such thatweight of every edge is we = £( ²2

ce )Any spanning tree T (with weights w) is (1+²)-spectrally thinOr, unweighted tree T is O(1/C )-spectrally thinThe same argument works if we replace ce by keand “spectrally thin” by “cut thin”.

weights u tree Tweights w

ke cut

cut

K cut

Uniformsampling

Cutsparsifier,

connectivityweights


nceweights

Karger

Unpublished


Spielman-Srivastava

Fung- Hariharan-

Harvey-Panigrahi


Asadpouret al.

Harvey-Olver

Linear-sizeSparsifiers

Batson-Spielman-Srivastava

Marcus-Spielman-Srivastava

?O(1)

thin trees

Corollary ofMSS

?Chekuri-Vondrak-

Zenklusen

Given a graph G with eff. conductances ¸ C.Find an unweighted spanning subtree T with

Easy lower bound: ® ¸ 1.5.Easy upper bound: ® = O(log n), algorithmic (even deterministic).

Main Theorem: ® = , algorithmic (even deterministic).

Theorem [MSS]: ® = O(1), existential result only.


Given an (unweighted) graph G with eff. conductances ¸ C.Can find an unweighted tree T with


Proof overview:1. Show independent sampling gives

spectral thinness, but not a tree.► Sample every edge e independently with

prob. xe=1/ce

2. Show dependent sampling gives a tree, and spectral thinness still works.

Matrix ConcentrationGiven any random nxn, symmetric matrices Y1,…,Ym.Is there an analog of Chernoff bound showing that i Yiis probably “close” to E[i Yi]?

Theorem: [Tropp ‘12]Let Y1,…,Ym be independent, PSD matrices of size nxn.Let Y=i Yi and Z=E [ Y ]. Suppose Yi ¹ R¢Z a.s. Then

Define sampling probabilities xe = 1/ce. It is known that e xe

= n–1.Claim: Independent sampling gives T µ E with E [|T|]=n–1 and

Theorem [Tropp ‘12]: Let M1,…,Mm be nxn PSD matrices.Let D(x) be a product distribution on {0,1}m with marginals x.Let Suppose Mi ¹ Z.ThenDefine Me = ce¢Le. Then Z = LG and Me ¹ Z holds.Setting ®=6 log n / log log n, we get whp.But T is not a tree!

Independent sampling

Laplacian of the single edge eProperties of conductances used



Proof overview:1. Show independent sampling gives spectral thinness,

but not a tree.► Sample every edge e independently with prob.

xe=1/ce

2. Show dependent sampling gives a tree, and spectral thinness still works.► Run pipage rounding to get tree T with Pr[ e2T ] = xe =

1/ce

Pipage rounding[Ageev-Svirideno ‘04, Srinivasan ‘01, Calinescu et al. ‘07, Chekuri et al. ‘09]

Let P be any matroid polytope.E.g., convex hull of characteristic vectors of spanning trees.Given fractional x

Find coordinates a and b s.t. linez x + z ( ea – eb ) stays in current faceFind two points where line leaves PRandomly choose one of thosepoints s.t. expectation is x

Repeat until x = ÂT is integral

x is a martingale: expectation of final ÂT is original fractional x.

ÂT1ÂT2

ÂT3

ÂT4

ÂT5

ÂT6

x

Say f : Rm ! R is concave under swaps if z ! f( x + z(ea-eb) ) is concave 8x2P, 8a, b2[m].Let X0 be initial point and ÂT be final point visited by pipage rounding.Claim: If f concave under swaps then E[f(ÂT)] · f(X0). [Jensen]

Let E µ {0,1}m be an event.Let g : [0,1]m ! R be a pessimistic estimator for E, i.e.,

Claim: Suppose g is concave under swaps. Then Pr[ ÂT 2 E ] · g(X0).

Pipage rounding and concavity

(e.g. f is multilinear extension of a supermodular function)

Chernoff BoundChernoff Bound: Fix any w, x 2 [0,1]m and let ¹ = wTx.Define . Then,

Claim: gt,µ is concave under swaps. [Elementary calculus]

Let X0 be initial point and ÂT be final point visited by pipage rounding.Let ¹ = wTX0. Then Bound achieved by independent sampling also achieved by pipage rounding

Matrix Pessimistic Estimators

Main Theorem: gt,µ is concave under swaps.

Theorem [Tropp ‘12]: Let M1,…,Mm be nxn PSD matrices.Let D(x) be a product distribution on {0,1}m with marginals x.Let Suppose Mi ¹ Z.LetThen and .

Bound achieved by independent sampling also achieved by pipage rounding

Pessimistic estimator



Proof overview:1. Show independent sampling gives spectral thinness,

but not a tree.► Sample every edge e independently with prob. xe=1/ce

2. Show dependent sampling gives a tree, and spectral thinness still works.► Run pipage rounding to get tree T with Pr[ e2T ] = xe =

1/ce

Matrix AnalysisMatrix concentration inequalities are usually proven via sophisticated inequalities in matrix analysisRudelson: non-commutative Khinchine inequalityAhlswede-Winter: Golden-Thompson inequalityif A, B symmetric, then tr(eA+B) · tr(eA eB).Tropp: Lieb’s concavity inequality [1973]if A, B symmetric and C is PD, then z ! tr exp( A + log(C+zB) ) is concave.Key technical result: new variant of Lieb’s theoremif A symmetric, B1, B2 are PSD, and C1, C2 are PD, then z ! tr exp( A + log(C1+zB1) + log(C2–zB2) ) is concave.

QuestionsO(1/C)-spectrally thin trees exist. Is

there an algorithm?Does sampling by edge connectivities give a cut sparsifierwith O(n log n) edges?Do O(1/K)-cut thin trees exist?

What about if we consider only the min cuts?

Do cut-sparsifiers with O(n²-2) edges exist for whichevery edge e has weight £(²2ke)?

Spectrally Thin Trees

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