Balasubramanian, R. and Subramanian, C.R. - On Sampling Colorings of Bipartite Graphs

dmtcs:368 - Discrete Mathematics & Theoretical Computer Science, January 1, 2006, Vol. 8
On Sampling Colorings of Bipartite Graphs

Authors: Balasubramanian, R. and Subramanian, C.R.

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, \emphalmost 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 \emphtorpid 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.


Volume: Vol. 8
Published on: January 1, 2006
Submitted on: March 26, 2015
Keywords: Graph colorings,Markov chains,Analysis of algorithms,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


Share

Consultation statistics

This page has been seen 99 times.
This article's PDF has been downloaded 81 times.