4 research outputs found
Random sampling of colourings of sparse random graphs with a constant number of colours
In this work we present a simple and efficient algorithm which, with high
probability, provides an almost uniform sample from the set of proper
k-colourings on an instance of a sparse random graph G(n,d/n), where k=k(d) is
a sufficiently large constant. Our algorithm is not based on the Markov Chain
Monte Carlo method (M.C.M.C.). Instead, we provide a novel proof of correctness
of our Algorithm that is based on interesting "spatial mixing" properties of
colourings of G(n,d/n). Our result improves upon previous results (based on
M.C.M.C.) that required a number of colours growing unboundedly with n.Comment: 30 pages 0 figures, uses fullpage.st
Deterministic counting of graph colourings using sequences of subgraphs
In this paper we propose a deterministic algorithm for approximately counting
the -colourings of sparse random graphs . In particular, our
algorithm computes in polynomial time a approximation of
the logarithm of the number of -colourings of for with high probability over the graph instances.
Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA
'06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation
decay} to compute {\em deterministically} marginals of {\em Gibbs
distribution}. We develop a scheme whose accuracy depends on {\em
non-reconstruction} of the colourings of , rather than {\em
uniqueness} that are required in previous works. This leaves open the
possibility for our schema to be sufficiently accurate even for .
The set up for establishing correlation decay is as follows: Given
, we alter the graph structure in some specific region of
the graph by deleting edges between vertices of . Then we show that
the effect of this change on the marginals of Gibbs distribution, diminishes as
we move away from . Our approach is novel and suggests a new context
for the study of deterministic counting algorithms