Approximately counting independent sets in bipartite graphs via graph containers

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to d-regular, bipartite graphs satisfying a weak expansion condition: when d is constant, and the graph is a bipartite omega(log(2)d/d)-expander, we obtain an FPTAS for the number of independent sets. Previously such a result ford > 5 was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a d-regular, bipartite alpha-expander, with alpha > 0 fixed, we give an FPTAS for the hard-core model partition function at fugacity lambda = omega(log d/d(1/4)). Finally we present an algorithm that applies to all dregular, bipartite graphs, runs in time exp( O( n . log(3)d /d to the number of independent sets. , and outputs a (1+o(1))-approximation