The Lopsided Lovász Local Lemma - · PDF fileTheorem (Erd}os, Lov asz 1975) ... Austin...

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The Lopsided Lov´ asz Local Lemma Austin Mohr Department of Mathematics Nebraska Wesleyan University With Linyuan Lu and L´ aszl´ o Sz´ ekely, University of South Carolina Austin Mohr The Lopsided Lov´ asz Local Lemma

Transcript of The Lopsided Lovász Local Lemma - · PDF fileTheorem (Erd}os, Lov asz 1975) ... Austin...

The Lopsided Lovasz Local Lemma

Austin Mohr

Department of MathematicsNebraska Wesleyan University

With Linyuan Lu and Laszlo Szekely, University of South Carolina

Austin Mohr The Lopsided Lovasz Local Lemma

Note on Probability Spaces

For this talk, every a probability space Ω is assumed to be uniformand equipped with the counting measure, so that

Pr (A) =|A||Ω|

for any subset A of Ω.

Austin Mohr The Lopsided Lovasz Local Lemma

Lovasz Local Lemma

Given a collection of mostly independent bad events, there is a wayto avoid them all.

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Hypergraph

Ground set V

Collection E of nonempty subsets of V

V = vi | i ∈ [5]E = v1, v2, v3, v2, v3, v3, v4, v5

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

A 2-coloring is an assignment of two colors to the vertices and isproper if no edge is monochromatic.

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

If there are many intersections among small edges, a hypergraphmay be impossible to 2-color properly.

How many intersections can we allow and still guarantee a proper2-coloring exists?

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Color the vertices of a hypergraph H with two colors independentlyat random.

For each edge e, define the “bad” event

Ae = 2-colorings of H | e is monochromatic.

The proper 2-colorability of H is equivalent to

Pr

∧e∈E(H)

Ae

> 0.

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Let e be an edge of the hypergraph H and F be a collection ofedges that are disjoint from e.

The event Ae is independent of the event algebra generated byAf | f ∈ F.

We capture this information in a dependency graph.

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Dependency Graph G [Erdos, Lovasz 1975]

Each vertex corresponds to an event.

Each event is independent of the event algebra generated byits non-neighbors in G .

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

For hypergraph 2-coloring, the graph G with

V (G ) = E (H) and

E (G ) = ef | e and f share a vertex in H

is a dependency graph.

Hypergraph to be 2-colored Dependency Graph

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Lemma (Symmetric Local Lemma - Erdos, Lovasz 1975)

Let Ai | i ∈ [n] be a collection of events having a dependencygraph G such that

G has maximum degree d and

Pr (Ai ) ≤ p for all i .

If ep(d + 1) ≤ 1, then

Pr

(n∧

i=1

Ai

)> 0.

Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

For the hypergraph 2-coloring problem,

Pr (Ai ) ≤ 22k

= p (k is the size of the smallest edge) and

d is the greatest number of intersections witnessed by anyedge.

The local lemma requires

ep(d + 1) ≤ 1

and so

d ≤ 2k−1

e− 1.

With d bounded, the local lemma concludes

Pr

(n∧

i=1

Ai

)> 0,

which means it is possible to properly 2-color the hypergraph.Austin Mohr The Lopsided Lovasz Local Lemma

2-Coloring Hypergraphs

Theorem (Erdos, Lovasz 1975)

Let H be a hypergraph in which every edge contains at least kvertices. If each edge intersects at most 2k−1

e − 1 other edges, thenH is properly 2-colorable.

Austin Mohr The Lopsided Lovasz Local Lemma

Lovasz Local Lemma

Lemma (Asymmetric Local Lemma - Spencer 1977)

Let Ai | i ∈ [n] be a collection of events having a dependencygraph G .If there are real numbers xi ∈ [0, 1) such that

Pr (Ai ) ≤ xi∏

ij∈E(G)

(1− xj)

for all i , then

Pr

(n∧

i=1

Ai

)≥

n∏i=1

(1− xi ) > 0.

The symmetric version follows from the asymmetric version bysetting each xi = 1

d+1 .

Austin Mohr The Lopsided Lovasz Local Lemma

Jealous?

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

A derangement is a permutation of [n] having no fixed point.

Define the canonical event

Ai = permutations π of [n] | π(i) = i .

The set⋂n

i=1 Ai contains precisely the derangements of [n].

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

The local lemma does not apply, since no pair of the events areindependent:

Pr (A1 ∧ A2) =(n − 2)!

n!=

1

n2 − n,

while

Pr (A1) Pr (A2) =(n − 1)!

n!· (n − 1)!

n!=

1

n2.

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

Fortunately, derangements possess a different useful property:

Pr (A1) =(n − 1)!

n!=

1

n

and

Pr(A1

∣∣ A2

)=|A1 ∩ A2||A2|

=(n − 1)!− (n − 2)!

n!− (n − 1)!=

1

n− 1

n(n − 1)2

soPr(A1

∣∣ A2

)≤ Pr (A1)

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

In fact,

Pr

(A1

∣∣∣∣∣k∧

i=2

Ai

)≤ Pr (A1)

for any k .

We capture this information in a negative dependency graph.

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

Negative Dependency Graph G [Erdos, Spencer 1991]

Each vertex corresponds to an event.

The inequality

Pr

Ai

∣∣∣∣∣∣∧j∈S

Aj

≤ Pr (Ai )

holds for each event Ai and any subset S of its non-neighborsin G .

Theorem (Lu, Szekely 2006)

The graph with vertex set [n] and no edges is a negativedependency graph for the canonical events Ai | i ∈ [n].

Austin Mohr The Lopsided Lovasz Local Lemma

Lopsided Local Lemma

Given a collection of mostly negative dependent bad events, thereis a way to avoid them all.

Lemma (Lopsided Local Lemma - Erdos, Spencer 1991)

Let Ai | i ∈ [n] be a collection of events having negativedependency graph G .If there are real numbers xi ∈ [0, 1) such that

Pr (Ai ) ≤ xi∏

ij∈E(G)

(1− xj)

for all i , then

Pr

(n∧

i=1

Ai

)≥

n∏i=1

(1− xi ) > 0.

Austin Mohr The Lopsided Lovasz Local Lemma

Derangements

Setting each xi = 1n , we verify

Pr (Ai ) ≤1

n

∏ij∈∅

(1− 1

n

)

and conclude

Pr

(n∧

i=1

Ai

)≥(

1− 1

n

)nn−→ 1

e.

Austin Mohr The Lopsided Lovasz Local Lemma

Hypergraph Matchings

The canonical event for a partial matching is the collection of allperfect matchings extending it.

partial matching B

canonical event AB

Austin Mohr The Lopsided Lovasz Local Lemma

Hypergraph Matchings

Conflict Graph

Each vertex corresponds to a partial matching.

Two matchings are adjacent if their union is not again apartial matching.

Austin Mohr The Lopsided Lovasz Local Lemma

Hypergraph Matchings

Theorem (Lu, M, Szekely 2013)

Let M be any collection of matchings in a complete uniformhypergraph. The conflict graph is a negative dependency graph forthe canonical events AM | M ∈M.

The set ⋂M∈M

AM

contains all perfect matchings of the complete uniform hypergraphthat extend no matching from M.

Austin Mohr The Lopsided Lovasz Local Lemma

Asymptotics from the Lopsided Local Lemma

ε-Near Positive Dependency Graph G [Lu, Szekely 2011]

Each vertex of G corresponds to an event.

Pr (Ai ∧ Aj) = 0 whenever ij ∈ E (G ).

The inequality

Pr

Ai

∣∣∣∣∣∣∧j∈S

Aj

≥ (1− ε) Pr (Ai )

holds for each event Ai and any subset S of its non-neighborsin G .

Theorem (M 2013)

Let M be a collection of matchings in a complete uniformhypergraph. If M is sufficiently “sparse”, then the conflict graphfor the canonical events AM | M ∈M is an ε-near positivedependency graph.

Austin Mohr The Lopsided Lovasz Local Lemma

Asymptotics from the Lopsided Local Lemma

Let A1, . . . , An be events in a probability space ΩN that growswith N and set µ =

∑ni=1 Pr (Ai ).

If the probabilities are appropriately controlled, then a negativedependency graph gives the lower bound

Pr

(n∧

i=1

Ai

)≥ (1− o(1))e−µ [Lu, Szekely 2011]

and a positive dependency graph gives the upper bound

Pr

(n∧

i=1

Ai

)≤ (1 + o(1))e−µ [M 2013]

as N tends to infinity.

Austin Mohr The Lopsided Lovasz Local Lemma

Asymptotics from the Lopsided Local Lemma

Corollary (M 2013)

Let A1, . . . , An be events in a probability space ΩN . If theconditions of the previous two theorems are satisfied, then∣∣∣∣∣

n⋂i=1

Ai

∣∣∣∣∣ = (1 + o(1))e−µ|ΩN |

as N tends to infinity.

Austin Mohr The Lopsided Lovasz Local Lemma

Counting Hypergraphs by Girth

A typical k-cycle in a hypergraph is one in which consecutiveedges intersect in exactly one vertex.

Figure : 3-cycle in 4-uniform hypergraph.

Austin Mohr The Lopsided Lovasz Local Lemma

Counting Hypergraphs by Girth

The configuration model [Bollobas 1980] connects matchingswith multihypergraphs.

2-Matching Multihypergraph

Figure : Configuration projecting to 3-regular, 2-uniform multihypergraphon four vertices.

Austin Mohr The Lopsided Lovasz Local Lemma

Counting Hypergraphs by Girth

Let M contain all matchings whose projection is a k-cycle withk < g .

For such a collection, the set ⋂M∈M

AM

contains precisely the matchings that represent hypergraphs ofgirth at least g in the configuration model.

Austin Mohr The Lopsided Lovasz Local Lemma

Counting Hypergraphs by Girth

Theorem (M, Szekely 2013+)

The number of r -regular, s-uniform hypergraphs having girth atleast g is

(1 + o(1))(rN)!

s!rN/s(rNs

)!(r !)N

exp

(−

g−1∑i=1

(r − 1)i (s − 1)i

2i

)

(assuming g, r , and s grow slowly with N).

Austin Mohr The Lopsided Lovasz Local Lemma

Homework

1 Think of your favorite combinatorial object.

Partial matchings

2 Define the canonical event for a particlar instance of thattype.

All perfect matchings extending it

3 Define conflict for two objects of that type.

Union is not a matching

4 Determine whether your conflict graph is a negativedependency graph.

Ask Laszlo

Austin Mohr The Lopsided Lovasz Local Lemma

Homework

1 Think of your favorite combinatorial object.

Partial matchings

2 Define the canonical event for a particlar instance of thattype.

All perfect matchings extending it

3 Define conflict for two objects of that type.

Union is not a matching

4 Determine whether your conflict graph is a negativedependency graph.

Ask Laszlo

Austin Mohr The Lopsided Lovasz Local Lemma

Homework

1 Think of your favorite combinatorial object.

Partial matchings

2 Define the canonical event for a particlar instance of thattype.

All perfect matchings extending it

3 Define conflict for two objects of that type.

Union is not a matching

4 Determine whether your conflict graph is a negativedependency graph.

Ask Laszlo

Austin Mohr The Lopsided Lovasz Local Lemma

Homework

1 Think of your favorite combinatorial object.

Partial matchings

2 Define the canonical event for a particlar instance of thattype.

All perfect matchings extending it

3 Define conflict for two objects of that type.

Union is not a matching

4 Determine whether your conflict graph is a negativedependency graph.

Ask Laszlo

Austin Mohr The Lopsided Lovasz Local Lemma

Thanks

Download This Talkwww.AustinMohr.com

Further Information

A. Mohr, Applications of the Lopsided Lovasz Local LemmaRegarding Hypergraphs, Ph.D. Dissertation (2013).

L. Lu, A. Mohr, and L. A. Szekely, Quest for NegativeDependency Graphs, Recent Advances in Harmonic Analysisand Applications (in honor of Konstantin Oskolkov), SpringerProceedings in Mathematics and Statistics 25 (2013).

L. Lu, A. Mohr, and L. A. Szekely, Connected BalancedSubgraphs in Random Regular Multigraphs Under theConfiguration Model , Journal of Combinatorial Mathematicsand Combinatorial Computing (2013+).

Austin Mohr The Lopsided Lovasz Local Lemma