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?
Efficient algorithm for coloring k-colorable graphs
Message passing and clustering (SAT)
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
1
2
3
4
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. 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 m=p¢choose(2,n) - 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:
based on analysis that uses partiallyrigorous tools
• All colorings within a
• Proved rigorously for k-SAT, k¸8• [AR06,MMZ05,MMZ05]
Every two clusters are “far”
cluster are “close”
• For k-SAT: not believed to be truefrom
for small
say k=3
each k,
other
• A linear number of
[MMW05]
• Exponentially many clusters
vertices are “frozen”
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 coloring
V1
V2
V3
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 plantedsolution 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.
Planted
Uniform
Density
< dk
Ck
Ck log n
Clustering phenomena
Survey Propagation
Alon and Kahale’s
coloring algorithm
[AK97]
Planted and Uniform
coincide
[AK97]
Clustering
Planted and Uniform “close”
Alon and Kahale’s coloring
algorithm [AK97]
Alon-Kahale’s coloring
algorithm [AK97]
What was known for SAT?
Dist.
Planted
Uniform
Density
> dk
Clustering phenomena
Survey Propagation [BMZ05]
Ck
Flaxman’s algorithm
[Fla03]
Version of k-opt
[FV04]
Ck log n
Planted and Uniform
coincide
Majority vote
Exponential time algorithm
[Chen03]
Planted and Uniform “close”
Flaxman, k-opt
Clustering
Majority vote works whp
[BBG02]
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
V2
V3
Proof Techniques – The Core
Every vertex is expected to have d/3 neighbors
in every other color class (d=np)
Claim 1: whp there d¸d
is no
subgraph H of G s.t.
0, d0 a sufficiently large constant
|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
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).
V1
V2
U - set of disagreeing vertices.
V3
(v)(v) ) v has d/3 neighbors in U.
|U|<n/100, |E(U)|>d|U|/6 – Contradicting Claim 2.
Proof Techniques – The Core
This is whp the case when np > Ck log n
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
V
V11
V
V22
V
V33
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:
There exists no
proper
planted
-exp{-C
1}n
Pr
[A] · e
3-coloring w.r.t
which there exists a big core
Claim 7:  · eexp{-C2}n, C2 > C1
Corollary: Pruniform[A] = o(1)
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
Outcome
differson
from
agrees
the
•Outcome
Core remains
colored
planted
on n/1000
Logarithmic
size
core
• Every colored
vertex
vertices
connected
components
agrees with planted
G[U] – graph induced by uncolored vertices
Exhaustively color G[U] according to G[V\U]
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
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 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 < 0 and zC < 0; 0 otherwise
xC = (x2 C’,C’C C’x) – (¬x2 C’’ C’’x)
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
p=d/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
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
Further Research
Loose
Rigorously analyze Survey Propagation on nearthreshold 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