Inapproximability of the independent set polynomial in the complex plane

Abstract

We study the complexity of approximating the value of the independent set polynomial ZG(λ ) of a graph G with maximum degree Δ when the activity λ is a complex number. When λ is real, the complexity picture is well understood, and is captured by two real-valued thresholds λ and λ c, which depend on Δ and satisfy 0 < λ < λ c. It is known that if λ is a real number in the interval ( λ , λ c) then there is a fully polynomial time approximation scheme (FPTAS) for approximating ZG(λ ) on graphs G with maximum degree at most Δ. On the other hand, if λ is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds λ and λ c on the Δ-regular tree. The "occupation ratio"of a Δ-regular tree T is the contribution to ZT (λ ) from independent sets containing the root of the tree, divided by ZT (λ ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if λ in [ λ , λ c]. Unsurprisingly, the case where λ is complex is more challenging. It is known that there is an FPTAS when λ is a complex number with norm at most λ and also when λ is in a small strip surrounding the real interval [0, λ c). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the complex values of λ for which the occupation ratio of the Δ-regular tree converges. These values carve a cardioid-shaped region λ Δ in the complex plane, whose boundary includes the critical points λ and λ c. Motivated by the picture in the real case, they asked whether λ Δ marks the true approximability threshold for general complex values λ . Our main result shows that for every λ outside of Λ Δ, the problem of approximating ZG(λ ) on graphs G with maximum degree at most Δ is indeed NP-hard. In fact, when λ is outside of λ Δ and is not a positive real number, we give the stronger result that approximating ZG(λ ) is actually #P-hard. Further, on the negative real axis, when λ < λ , we show that it is #P-hard to even decide whether ZG(λ ) > 0, resolving in the affirmative a conjecture of Harvey, Srivava, and Vondr'ak. Our proof techniques are based around tools from complex analysis-specifically the study of iterative multivariate rational maps.