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}
1
and σ(v)≠σ(u) for every {v,u}E
1
1
2
1
3
Colours : {1, 2, 3, 4}
6
3
2
5
3
4
Counting Vs Sampling
G=(V, E)
Set of colours {1,…,k}
Z(G, k):  k-colouring of G
Process: Choose u.a.r. a k-colouring of the graph
Ei,j: “Vertices i, j receive different colour”
2
1
c
d
a
3
e
6
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
b
4
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 – arbitrary graph
[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
 Bandyopadhyay, Gamarnik 2005
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)
Efficient computation of
marginals – Correlation decay
u
v
A
B
t
C
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.
Reduce computational load
differently…
u
l1
r1
l2
r2
l3
r3
l4
r4
v
A
t
B
C
Reduce computational load
differently…
u
l1
r1
l2
r2
l3
r3
l4
r4
v
A
B
C
Implications on spatial mixing
conditions
u
l1
r1
l2
r2
l3
r3
l4
r4
v
A
t
L(l3, t): Vertices outside red cycle
B
C
t
L(r3, t): Vertices outside green cycle
Comparison with the first
approach
u
l1
r1
l2
r2
l3
r3
l4
r4
v
t
L(l3, t): Vertices outside red cycle
Theorem - Accuracy
u
l1
r1
l2
r2
l3
r3
l4
r4
v
Spatial Correlation decay
G=(V,E)
u
BP(v,u)
Product measure: Pq
Pr[Disagreeing]=q
Pr[Non-Disagreeing]=1-q
v
“Path of disagreement
between u & v”
Bounding Spatial Correlation
decay
:
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-dense of 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.
Thank You!