Why almost all kcolorable graphs are easy

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Why almost all k-colorable graphs are easy ?

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

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Talk Outline

• Random graphs phase transitions and clustering
• How do typical k-colorable graphs look?
• Efficient algorithm for coloring k-colorable
  graphs
• Message passing and clustering (SAT)

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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

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The k-Coloring Problem

• Finding a 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.
  pp(n)
• ?(Gn,p )np/2ln(np) for np2c0,n/log7n
  Bol88,Luc91

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Phase transitions and clustering

• Consider the variant Gn,m of Gn,p
• Choose uniformly at random mm(n) edges
• When mpchoose(2,n) - Gn,m and Gn,p are close
• There exists a constant dd(k) such that
• 2m/ngtd almost all graphs in Gn,m are not
  k-colorable
• 2m/nltd almost all graphs are k-colorable Fri99
• Such phenomena is called a phase transition

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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

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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
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Phase transitions and clustering

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

based on analysis that uses partially-rigorous
tools

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

• Proved rigorously for k-SAT, k8
  AR06,MMZ05,MMZ05
• For k-SAT not believed to be true for small k,
  say k3
• MMW05

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

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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

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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 coloring

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Our Results

• Characterization of Gn,m k colorability
• 2m/nCk, 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

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Our Results

• There exists a deterministic polynomial time
  algorithm that k-colors almost all k-colorable
  graphs with mgtCkn 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 !
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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-Kahales coloring algorithm AK97 works for
  Gn,m k colorability as well
• Gn,m,k also has the same clustering structure

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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
• Flaxmans 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

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What was known so far?
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What was known for SAT?
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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 (k3)

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Proof Techniques The Core

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

Claim 1 whp there is no subgraph H of G s.t.
V(H)ltn/100 and E(H)gtdH/10
dd0, d0 a sufficiently large constant
Claim 2 whp there are no two proper 3-colorings
at distance greater than n/100
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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(?,?)ltn/100 (Claim 1). U - set of
disagreeing vertices. ?(v)??(v) ) v has d/3
neighbors in U. Ultn/100, E(U)gtdU/6
Contradicting Claim 2.
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Proof Techniques The Core

• This is whp the case when np gt Ck log n
• When npO(1) whp not the case

• Definition of Core H v2H if
• v has at least np/4 neighbors in GH 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

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Proof Techniques The Core

• Corollary
• (1-exp-?(np))n vertices are frozen in every
  proper 3-coloring
• Only one cluster of exponential size

V1
V2
V3
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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 PruniformA ?PrplantedA
Intuition typically there are at most ? ways to
generate G in the planted model. Now use a union
bound.
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Moving to the Uniform Case

• A the graph has no big core

Claim 6 PrplantedA e-exp-C1n
There exists no proper 3-coloring w.r.t which
there exists a big core
Claim 7 ? eexp-C2n, C2 gt C1
Corollary PruniformA o(1)
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Algorithmic Perspective

• Show that Alon and Kahales algorithm AK97
  works in the uniform case
• What is Alon and Kahales algorithm?
• Approximate a proper 3-coloring (spectral
  techniques)
• Refine the coloring recoloring step
• Uncolor suspicious vertices
• GU graph induced by uncolored vertices
• Exhaustively color GU according to GV\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
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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 GV \ 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

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Algorithmic Perspective - SAT

• Show that Flaxmans algorithm Fla03 works in
  the uniform case
• What is Flaxmans algorithm?
• Approximate a satisfying assignment (majority
  vote)
• Unassign suspicious variables
• GU graph induced by unassigned variables
• Exhaustively satisfy GU according to GV \ U

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SAT and Message Passing

• Warning Propagation
• Given a formula 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)
• C?x 1 if y?C lt 0 and z?C lt 0 0 otherwise
• x?C (?x2 C,C?C C?x) (?x2 C C?x)

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SAT and Message Passing

• WP(F)
• Repeat until no message changes
• Initialize all messages C?x to 1/0 w.p. 0.5
• Randomly order the edges of G(F)
• Evaluate all messages C?x
• Assign every x according to (?x2 C C?x) (?x2
  C C?x)

• Theorem FMV06 If F sampled according to
  Planted 3SAT
• pd/n2, d sufficiently large constant, then whp
• WP converges after O(log n) iterations
• Assigned variables agree with some satisfying
  assignment
• All but exp-?(d)n variables are assigned
• Clauses of unassigned variables are easy to
  satisfy

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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

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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

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