DMTCS

Volume 8

n° 1 (2006), pp. 17-30

author:R. Balasubramanian and C.R. Subramanian
title:On Sampling Colorings of Bipartite Graphs
keywords:Graph colorings, Markov chains, Analysis of algorithms
abstract:We study the problem of efficiently sampling
k
-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any
k
,
6 ≤ k ≤ n
{1/3-ε}
,
ε > 0
fixed, almost every bipartite graph on
n+n
vertices is such that the mixing time of any markov chain asymptotically uniform on its
k
-colorings is exponential in
n/k
2
(if it is allowed to only change the colors of
O(n/k)
vertices in a single transition step). This kind of exponential time mixing is called torpid mixing. As a corollary, we show that there are (for every
n
) bipartite graphs on
2n
vertices with
Δ(G) = Ω(
ln
n)
such that for every
k
,
6 ≤ k ≤ Δ/(6
ln
Δ)
, each member of a large class of chains mixes torpidly. While, for fixed
k
, such negative results are implied by the work of CDF, our results are more general in that they allow
k
to grow with
n
. We also show that these negative results hold true for
H
-colorings of bipartite graphs provided
H
contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (
k
-colorings or
H
-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed
k
or fixed
H
, such negative results are implied by the work of CDF, our results are more general in that they allow
k
or
H
to vary with
n
.
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reference: R. Balasubramanian and C.R. Subramanian (2006), On Sampling Colorings of Bipartite Graphs, Discrete Mathematics and Theoretical Computer Science 8, pp. 17-30
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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