On Sampling Colorings of Bipartite Graphs - R. Balasubramanian; C. R. Subramanian (Discrete Mathematics & Theoretical Computer Science, 2006)
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.