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 (if it is allowed to only change the colors of 2 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( such that for every
ln n)k , 6 ≤ k ≤ Δ/(6 ,
each member of a large class of chains mixes torpidly.
While, for fixed ln Δ)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 . |
| If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. | |
| 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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Automatically produced on Thu Feb 9 13:05:17 CET 2006 by falk