Approximately counting independent sets in bipartite graphs via graph containers
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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 Ω( log^2 d/d)-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for d>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 α-expander, with α>0 fixed, we give an FPTAS for the hard-core model partition function at fugacity λ=Ω(log d / d^1/4). Finally we present an algorithm that applies to all d-regular, bipartite graphs, runs in time exp( O( n ·log^3 d /d ) ), and outputs a (1 + o(1))-approximation to the number of independent sets.
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