Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model

Abstract

Abstract. We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree ∆. More generally, for an input graph G = (V, E) and an activity λ > 0, we are interested in the quantity ZG(λ) defined as the sum over independent sets I weighted as w(I) = λ |I| . In statistical physics, ZG(λ) is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negibile 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 ∆ when ∆ is constant and λ < λc(T∆) := (∆ − 1) ∆−1 /(∆ − 2) ∆ . The quantity λc(T∆) is the critical point for the so-called uniqueness threshold on the infinite, regular tree of degree ∆. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for λc(T∆) < λ < λc(T∆) + ε(∆), unless NP=RP, for some function ε(∆) > 0. We remove the upper bound in the assumptions of Sly's result for ∆ = 4, 5, that is, we show that there does not exist efficient randomized approximation algorithms for all λ > λc(T∆) for ∆ = 3 and ∆ ≥ 6. 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 ∆ 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.

Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

“…We will prove here that the convergence of BP provides the existence of a distance function Φ satisfying (7). We defer the technical proof of (6) to Section A of the appendix. The Jacobian J of the BP operator F is given by…”

“…On the other side, Sly [33] (extended in [6,7,34,8]) has established that, unless N P = RP , for all ∆ ≥ 3, there exists γ > 0, for all λ > λ c (∆), there is no polynomial-time algorithm for triangle-free ∆-regular graphs to approximate the partition function within a factor 2 γn .…”

The Ising Partition Function: Zeros and Deterministic Approximation

“…For a more detailed account of this phenomenon in the context of partition functions, see, e.g., [46, Appendix A]. 1 the partition function is NP-hard to approximate outside this regime [19,49]. Perhaps surprisingly, however, no deterministic approximation algorithm is known for the classical ferromagnetic Ising partition function that works over anything close to the full range of the randomized algorithm of [27]: to the best of our knowledge, the best deterministic algorithm, due to Zhang, Liang and Bai [56], is based on correlation decay and is applicable to graphs of maximum degree ∆ only when β > 1 − 2/∆.…”

“…Thus, by eq. (19) and continuity, we can take z 2 large enough in absolute value such that z 1 as de ned in eq. (18) lies on the unit circle.…”

Counting independent sets in unbalanced bipartite graphs

“…For graphs of maximum degree ∆, there is an FPTAS due to Weitz [27] when λ < λ c (∆), the uniqueness threshold for the infinite ∆-regular tree. On the other hand, Sly [25], Sly and Sun [26], and Galanis, Stefankovic, and Vigoda [11] showed that there is no polynomial-time approximation algorithm for λ > λ c (∆) unless NP = RP.…”