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Improved inapproximability results for counting independent sets in the hard-core model


  • Supported by NSF (CCF-0830298, CCF-0910584 and CCF-121745B) (A.G., E.V., L.Y.); NSF (CCF-0910415) (Q.C., D.S.).


We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph math formula and an activity math formula, we are interested in the quantity math formula defined as the sum over independent sets I weighted as math formula. In statistical physics, math formula is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and math formula. The quantity math formula is the critical point for the so-called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for math formula, unless math formula, for some function math formula. We remove the upper bound in the assumptions of Sly's result for math formula, that is, we show that there does not exist efficient randomized approximation algorithms for all math formula for math formula and math formula. Sly's inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014