Approximately counting independent sets in bipartite graphs via graph containers

Abstract

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.

Enumerating independent sets in Abelian Cayley graphs

“…The ideas introduced in this method have proved to be extremely useful, finding a number of applications in combinatorics. upper bound on phase transition on the hardcore model on Z d [GK04], lower bounds for mixing for Glauber dynamics for hardcore model in bipartite regular graphs [GT06], enumerating uniform intersecting set systems [BGLW21], enumerating q-colorings of the discrete torus [Gal03], [KJ20], [JK20], phase coexistence of the 3-coloring model in Z d [GKRS15], more detailed descriptions of independent sets in the hypercube [BGL21], [Gal10], [JP20], [JPP21b], [KP19], [Par21], and faster algorithms for approximately counting independent sets in bipartite graphs [JPP21a].…”