Why almost all k-colorable graphs are easy

A. Coja-Oghlan, M. Krivelevich, D. Vilenchik

Talk Outline

Random graphs: phase transitions and clustering

How do typical k-colorable graphs look?

Message passing and clustering (SAT)

Proof techniques – two approaches

The k-Coloring Problem

Given a graph G=(V,E):

Find f : V![k] s.t. 8(u,v)2E(G): f(u)f (v)

Find f with minimal possible k

Such k is called the chromatic number of G, (G)

E.g. (G)=3

3 4

1 2

The k-Coloring Problem

Finding a proper k-coloring is NP Hard

No polynomial time algorithm approximates (G) within factor better than n1- (unless NPµZPP) [FK98]

How to proceed? random models and average case analysis

Gn,p - every possible edge is included w.p. p=p(n)

(Gn,p)=np/2ln(np) for np2[c0,n/log7n] [Bol88,Luc91]

Phase transitions and clustering

Consider the variant Gn,m of Gn,p :

Choose uniformly at random m=m(n) edges

When , Gn,m and Gn,p are “close”

There exists a constant d=d(k) such that

2m/n>d: almost all graphs in Gn,m are not k-colorable

2m/n<d: almost all graphs are k-colorable [Fri99]

Such phenomena is called a phase transition

Phase transitions and clustering

Gn,m with 2m/n just below the threshold is “hard” experimentally

Possible explanation (partially non-rigorous) comes from statistical physics [MPWZ02]

The “geometrical” structure of the space of proper k-colorings - the clustering phenomena

Need to define notion of distance

Phase transitions and clustering

Two k-colorings are the same if they differ only by a permutation of the color classes

Two k-colorings , are at distance t if

they disagree on the color of at least t vertices in every permutation of the color classes.

There exists one permutation obtaining equality

Similar to Hamming distance

Phase transitions and clustering

Gn,m with 2m/n just below the threshold:

• All colorings within a cluster are “close”• A linear number of vertices are “frozen”

• Every two clusters are “far” from each other• Exponentially many clusters

based on analysis that uses partially-rigorous tools

• Proved rigorously for k-SAT, k¸8 [AR06,MMZ05]• For k-SAT: not believed to be true for small k, say k=3 [MMW05]

Phase transitions and clustering

Why does this structure make life hard?

Heuristics get “distracted” by this structure

Every cluster “pulls” in its direction

Heuristics try to find a compromise between clusters

This is impossible due to the structure

Survey Propagation does well in practice [BMWZ05]

Random k-colorable graphs

Gn,m with 2m/n above the threshold – not suitable to study k-colorable graphs

Instead, consider Gn,m | { k-colorability }

The uniform distribution over k-colorable graphs with exactly m edges

Another possibility, the planted model Gn,m,k

Partition the vertex set into k color classes of size n/k

Include m random edges that respect the coloringV1

V3

V2

Our Results

Characterization of Gn,m | { k –colorability }

2m/n=Ck, Ck a sufficiently large constant

Using rigorous analysis we show that typically:• Single cluster of proper k-colorings• Size of the cluster is exponential in n

• (1-exp{-(Ck)})n vertices are “frozen”

Our Results

There exists a deterministic polynomial time algorithm that k-colors almost all k-colorable graphs with m>Ckn edges. Ck a sufficiently large constant.

Rigorously complement results for sparse case:When clustering is simple – the problem is easy

When clustering is “complicated” – the problem is harder (?)

Almost all k-colorable graphs are easy!

Our Results

Show that Gn,m,k and Gn,m | { k –colorability } share many structural properties (“close”)

Justifying the somewhat unnatural usage of planted-solution models

Alon-Kahale’s coloring algorithm [AK97] works for Gn,m | { k –colorability } as well

Gn,m,k also has the same clustering structure

Our Results

Our results also apply to the k-SAT setting

Similar threshold and clustering phenomena are known/believed for k-SAT

The planted and uniform SAT distributions are “close”

Flaxman’s algorithm for planted 3CNF formulas works for the uniform setting

Improving the exponential time algorithm for uniform satisfiable 3CNFs (only one known so far)

Answering open research questions in [BBG02]

What was known so far?Dist.

Density

PlantedUniform

>dk Clustering phenomena

Survey Propagation

Ck Alon and Kahale’s

coloring algorithm [AK97]

Clustering Planted and Uniform “close” Alon and Kahale’s coloring

algorithm [AK97]

Ck log n Planted and Uniform coincide

[AK97]

Alon-Kahale’s coloring algorithm [AK97]

What was known for SAT?Dist.

Density

PlantedUniform

<dk Clustering phenomena

Survey Propagation [BMZ05]

Ck Flaxman’s algorithm

[Fla03] Version of k-opt

[FV04]

Exponential time algorithm [Chen03]

Planted and Uniform “close”

Flaxman, k-opt

Clustering

Ck log n Planted and Uniform coincide

Majority vote

Majority vote works whp [BBG02]

Algorithmic Perspective

Show that Alon and Kahale’s algorithm [AK97] works in the uniform case

What is Alon and Kahale’s algorithm?

Approximate a proper 3-coloring (spectral techniques)

Refine the coloring – recoloring step

Uncolor “suspicious” vertices

G[U] – graph induced by uncolored vertices

Exhaustively color G[U] according to G[V \U]

Outcome differs from planted on n/1000

vertices

Outcome agrees on the core

• Core remains colored• Every colored vertex agrees with planted

Logarithmic size connected components

Algorithmic Perspective - SAT

Show that Flaxman’s algorithm [Fla03] works in the uniform case

What is Flaxman’s algorithm?

Approximate a satisfying assignment (majority vote)

Unassign “suspicious” variables

G[U] – graph induced by unassigned variables

Exhaustively satisfy G[U] according to G[V \U]

SAT and Message Passing

Warning Propagation:

Given a 3CNF F – define Factor Graph G(F)

Bipartite graph: V1 = variables, V2 = clauses

(x,C)2E(G) iff x appears in C

Two types of messages: C=(xÇyÇz)

Cx = 1 if yC < 0 and zC < 0; 0 otherwise

xC = (x2 C’,C’C C’x) – (¬x2 C’’C’’x)

SAT and Message Passing

WP(F)

Initialize all messages Cx to 1/0 w.p. 0.5

Repeat until no message changes:

• Randomly order the edges of G(F)

• Evaluate all messages Cx

Assign every x according to (x2C’ C’x) – (¬x2C’’ C’’x)

Theorem [FMV06]: If F sampled according to Planted 3SAT p=d/n2, d sufficiently large constant, then whp:

• WP converges after O(logn) iterations• Assigned variables agree with some satisfying assignment• All but exp{-(d)}n variables are assigned• Clauses of unassigned variables are “easy” to satisfy

SAT and Message Passing

Our work implies – [FMV06] applies for the uniform SAT setting as well

Reinforces the following thesis:

When clustering is complicated ) formulas are hard ) sophisticated algorithms needed: Survey Propagation

When clustering is simple ) formulas are easy ) naïve algorithms work: Warning Propagation

Clustering: Proof Techniques

Recall, Gn,m | { k-colorability }

The uniform distribution over k-colorable graphs with exactly m edges

Why more difficult than the planted distribution?

Edges are not independent

For starters, consider the planted distribution Gn,p,k (k=3) V1

V3

V2

Proof Techniques – The Core

Every vertex is expected to have d/3 neighbors in every other color class (d=np)

Claim 1: whp there is no subgraph H of G s.t. |V(H)|<n/100 and E(H)>d|H|/10

Claim 2: whp there are no two proper 3-colorings at distance greater than n/100

d¸d0, d0 a sufficiently large constant

Proof Techniques – The Core

Claim 3: Suppose that every vertex has the expected degree, and Claims 1 and 2 hold. Then the graph G is uniquely 3-colorable.

Proof: - the planted coloring. If not unique, 9, dist(,)<n/100 (Claim 1).U - set of disagreeing vertices.v)(v) ) v has d/3 neighbors in U.|U|<n/100, |E(U)|>d|U|/6 – Contradicting Claim 2.

V1

V3

V2

Proof Techniques – The CoreThis is whp the case when np > Ck logn

When np =O(1) – whp not the case

Definition of Core H : v2H if v has at least np/4 neighbors in G[H] in

every other color class

v has at most np/10 neighbors outside of H.

Claim 4: 9 Core H s.t. whp

|H | ¸ (1-exp{-(np)})n H is uniquely 3-colorable

Proof Techniques – The Core

Corollary:

(1-exp{-(np)})n vertices are frozen in every proper 3-coloring

Only one cluster of exponential size

V1

V3

V2V1 V2

V3

Moving to the Uniform Case

A – a “bad” graph property (e.g. the graph has no big core)

– the expected number of proper k-colorings of random graph in the planted distribution

Claim 5: Pruniform[A] · ¢Prplanted[A]

Intuition: typically there are at most ways to generate G in the planted model. Now use a union bound.

Moving to the Uniform Case

A – “the graph has no big core”

Claim 6: Prplanted[A] · e-exp{-C1}n

Claim 7: · eexp{-C2}n, C2 > C1

Corollary: Pruniform[A] = o(1)

There exists no proper 3-coloring w.r.t which there exists a big core

Algorithmic Perspective - Analysis

Typically, uniform graphs have a big core

Two more facts needed for the analysis:

Claim 1 in the uniform case

Logarithmic size components in G[V \H]

Both properties hold w.p. 1-1/poly(n) in the planted model - cannot use “union bound”

Solution: analyze directly the uniform distribution

Difficulty: edges are strongly dependent

Solution: careful, non-trivial, counting argument

Proof Techniques – a sampler

Proof Technique - using the union bound:

Fix a set U of t vertices, t·n/100

Fix a “bad” graph on the vertices in U

Upper bound the probability of such graph

Sum over all possible choices for U

Claim 1: whp there is no subgraph H of G s.t. |V(H)|<n/100 and E(H)>C|H|/10

Gn,m | { k –colorability }, m=Cn, C

some large constant.

Proof Techniques – a sampler

There are ways to choose the set U

There are ways to fix the “bad” graph

Pr[ the bad graph is a subgraph of G ] =

If G was in Gn,p then = pdt/10

Recall Gn,m | { k –colorability }

In the uniform distribution, what is p ?

Proof Techniques – a sampler

Let p be s.t.

We shall Prove:

Pruniform[ fixed subgraph with r edges ] · (6p)r

Assume we did, then the probability of the claim

not holding is at most

The expected number of edges had the graph been sampled according to the

planted distribution

p=3m/n2

Proof Techniques – a sampler

Our goal: prove that

Pruniform[ fixed subgraph with r edges ] · (6p)r

Fix “bad” subgrpah B

V1=Gi contains B as subgraph V2=all k-colorable graphs with m edges

G1

G2

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.

H1

H2

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.

.

Proof Techniques – a sampler

1– average degree of left side

2– average degree of right side

Double counting: |V1|1 = |V2|2

|V1| · |V2|(21)

We show a bipartite graph s.t. (21)=(6p)r

(Gi,Hj)2 E if Gi can be obtained from Hj using the following procedure:

G1

G2

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.

.

H1

H2

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.

Proof Techniques – a sampler

Remove edges of B from Gi, and replace them arbitrarily s.t. at least one k-coloring is respected

Denote

Observe that:

Therefore,

Further Research

Loose

Rigorously analyze Survey Propagation on near-threshold formulas/graphs

First step – analyze Survey Propagation on Planted instances

Prove the near-threshold clustering phenomena

Rigorously analyze message passing algorithms

Analyze instances with an arbitrary constant (above the threshold) density