Inapproximability of the Independent Set Polynomial Below the Shearer Threshold

We study the problem of approximately evaluating the independent set polynomial of boundeddegree graphs at a point λ. Equivalently, this problem can be reformulated as the problem of approximating the partition function of the hard-core model with activity λ on graphs G of maximum degree ∆. For λ > 0, breakthrough results of Weitz and Sly established a computational transition from easy to hard at λc(∆) = (∆ − 1) /(∆ − 2), which coincides with the tree uniqueness phase transition from statistical physics. For λ < 0, the evaluation of the independent set polynomial is connected to the problem of checking the conditions of the Lovász Local lemma (LLL) and applying its algorithmic consequences. Shearer described the optimal conditions for the LLL and identified the threshold λ(∆) = (∆ − 1)/∆ as the maximum value p such that every family of events with failure probability at most p and whose dependency graph has maximum degree ∆ has nonempty intersection. Very recently, Patel and Regts, and Harvey et al. have independently designed FPTASes for approximately computing the partition function whenever |λ| < λ(∆). Our main result establishes for the first time a computational transition at the Shearer threshold. Namely, we show that for all ∆ ≥ 3, for all λ < −λ(∆), it is NP-hard to approximate the partition function on graphs of maximum degree ∆, even within an exponential factor. Thus, our result, combined with the algorithmic results for λ > −λ(∆), establishes a phase transition for negative activities. In fact, we now have a complete picture for the complexity of approximating the partition function for all λ ∈ R and all ∆ ≥ 3, apart from the critical values. 1. For −λ(∆) < λ < λc(∆), there exists an FPTAS for approximating the partition function with activity λ on graphs G of maximum degree ∆. 2. For λ < −λ(∆) or λ > λc(∆), it is NP-hard to approximate the partition function with activity λ on graphs G of maximum degree ∆, even within an exponential factor. Rather than the tree uniqueness threshold of the positive case, the phase transition for negative activities corresponds to the existence of zeros for the partition function of the tree below −λ(∆).