Counting independent sets in graphs with bounded bipartite pathwidth

Abstract

We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalization of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw‐free graphs, which generalize line graphs. We consider two further generalizations of claw‐free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially bounded vertex weights.

Counting Weighted Independent Sets beyond the Permanent

“…has only real negative roots when G is claw-free. This was proved in [8] for unit weights, and extended to general weights in [12]. However, the algorithm requires the reduction from approximate counting to approximate random generation [24].…”

“…The freedom to use very large vertex weights allows us to approximate W k (G) for any 0 ≤ k ≤ α(G). For k < α(G), we would need to use the algorithm described in [12], which is an improvement of the approach of that in [22]. This method is based on the fact that the independence polynomial…”

“…Secondly, the process used in section 2.2, replacing subgraphs by revised weights on vertices, makes it difficult to track W k (G) efficiently by combining contributions from different modified vertices. For polynomially-bounded weights, this is possible by a dynamic programming approach, but this case is resolved in [12] for all clawfree graphs. However, for weights exponential in n, there can be an exponential number of cases, which must be combined efficiently somehow.…”

“…So what do we achieve by restricting to (claw, odd hole)-free graphs? The gain is that our algorithm is genuinely polynomial time, whereas that of [12] is only pseudopolynomial. In particular, this allows us to approximate the total weight of independent sets of any given size k. We can estimate the total weight of maximum independent sets, which corresponds to counting maximum matchings in the root graph of a line graph.…”

“…Observe that the algorithm of [12] runs in polynomial time. It counts all weighted independent sets in an arbitrary claw-free graph G = (V, E) with unary weights.…”