Counting Graph Colourings by using Sequences of Subgraphs

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Counting Graph Colourings by using Sequences of Subgraphs. Charilaos Efthymiou DIMAP University of Warwick DIMAP Summer School – July 2010. Counting. Problem : Given Find the cardinality of the set of feasible solutions

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## Counting Graph Colourings by using Sequences of Subgraphs

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### Counting Graph Colourings by using Sequences of Subgraphs

Charilaos Efthymiou

DIMAP

University of Warwick

DIMAP Summer School – July 2010

Counting
• Problem: Given

Find the cardinality of the set of feasible solutions

• Examples: matching, independent sets, proper colourings, bin packing, SAT
Graph Colouring

G=(V, E)

Set of colours{1,…,k}

(Proper) Colouring σ:V →{1,…,k}

and σ(v)≠σ(u)for every {v,u}E

1

1

2

1

1

3

Colours :{1, 2, 3, 4}

6

3

2

4

3

5

Counting Vs Sampling

G=(V, E)

2

1

Set of colours{1,…,k}

c

d

Z(G, k):  k-colouring of G

a

3

e

Process: Choose u.a.r. a k-colouring of the graph

b

6

Ei,j: “Vertices i, j receive different colour”

4

5

pa:: Pr[ E1,6 in Ga]

pb: Pr[ E1,4 in Gb]

pc: Pr[ E2,6 in Gc]

pd: Pr[ E2,5 in Gd]

pe: Pr[ {6,3} in Ge]

Z(G,k)=|V|k papbpcpdpe

Counting
• Exact counting is hard,
• Valiant ‘79P class
• Approximate counting using

Rapidly Mixing Markov Chains

• Celebrated achievements: FPRAS for
• Permanents: Jerrum, Sinclair 1989, Jerrum Sinclair Vigoda 2001
• Volume of convex body: Dyer, Frieze Kannan 1991
• Counting independent sets in degree-4 graphs: Luby and Vigoda 1997
Graph k-colourings
• Maximum degree Δ
• [Vigoda 99] k>11/6Δ– arbitrarygraph
• [Mοssel & Sly 08] Random graphs with fixed expected degree d , with k>f(d)
• [Hayes, Vera & Vigoda]

Planar Graphs k> Ω(Δ/log Δ)

For this talk…
• We propose algorithms which are not based on Markov Chains.
• Compute the corresponding probabilities directly.
• Weakness:
• We only compute ε-approximation to log Z(G, k)
• Strength:
• Deterministic
• Explicit results for Gnp
Works for Det. Counting
• Colourings
• Regular graphs with high girth Δ+1-colours- PTAS
• G with girth 4, 2.8Δ-colours FPTAS
• Gamarnik, Katz 2007
• Sparse Random Graphs with number of colours that depend on the expected degree -PTAS
• Efthymiou, Spirakis 2008.
• Graph matchings - FPTAS
• Bayatui, Gamarnik, Katz, Nair, Tetali 2007.
• Independent sets FPTAS
• Weitz 2006
Easy Examples…
• Trees
• Compute each probability recursively!
• DP - - For constant k the time-complexity is O(n)
• Graphs of bounded treewidth
• Graphs with number of k-colourings (proper & non-proper) that is O(nc)
3-step algorithm
• Compute Pr[Euv] on the “small” graph.
• Prove independence from boundary conditions.
• Dobrushin’s Condition for Uniqueness of Gibbs measure
• Project to the initial graph.
Implications on spatial mixing conditions

l1

r1

u

l2

r2

v

l3

r3

l4

r4

A

B

C

t

t

L(l3, t): Vertices outside red cycle

L(r3, t): Vertices outside green cycle

Comparison with the first approach

l1

r1

u

l2

r2

v

l3

r3

l4

r4

t

L(l3, t): Vertices outside red cycle

Theorem - Accuracy

l1

r1

u

l2

r2

v

l3

r3

l4

r4

Spatial Correlation decay

G=(V,E)

u

BP(v,u)

Product measure: Pq

Pr[Disagreeing]=q

Pr[Non-Disagreeing]=1-q

“Path of disagreement between u & v”

v

Applications I – Sparse Gnp
• The underlying graph is Gnp with expected degree d, d is fixed
• Vertex set V={1,…, n}
• Each possible edge appear with probability p, independently of the others.
• Expected degree is d is fixed real, i.e. p=d/n
• The maximum degree is Θ(log n/ loglog n)
• Chromatic number Constant
Applications I – Sparse Gnp
• Isolate Θ(log n) neighborhoods around u,v
• Tree with additional Θ(log n) edges
• Computations by Dynamic Programming
• Using k≥(2+ε)d with probability 1-n-Ω(1) we get a polynomial time, n-Ω(1)-approximation of log Z(Gnp,k).
Applications II – Locally α-dense graphs
• G(V,E) is locally α-denseof bounded maximum degree Δ if
• For all {w1, w2} E w2 has at most (1-α)Δ neighbors which are not adjacent to w1
• α  [0,1] is a parameter of the model
Applications II – Locally dense graphs
• For k>(2-α)Δ we get a (log n)-Ω(1)-approximation of log Z(G,k), in polynomial time.
• If, additionally, every Θ(log n) neighborhood of G has constant treewidth, then we get a n-Ω(1)-approximation of log Z(G,k), in polynomial time.