Inapproximability of the independent set polynomial in the complex plane

We study the complexity of approximating the value of the independent set polynomial Z G (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 * c . It is known that if λ is a real number in the interval (−λ * ,λ c ) then there is an FPTAS for approximating Z G (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 Z T (λ) from independent sets containing the root of the tree, divided by Z T (λ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if λ∈ [−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 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 Z G (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 Z G (λ) is actually #P-hard. Further, on the negative real axis, when λ * , we show that it is #P-hard to even decide whether Z G (λ)u003e0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps.