Inapproximability of the independent set polynomial in the complex plane

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 λc, which depend on Δ and satisfy 0<λ*<λc. It is known that if λ is a real number in the interval (−λ*,λc) then there is an 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 λc 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∈ [−λ*,λc]. 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,λc). 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 ΛΔ in the complex plane, whose boundary includes the critical points −λ* and λc. Motivated by the picture in the real case, they asked whether ΛΔ marks the true approximability threshold for general complex values λ. Our main result shows that for every λ outside of ΛΔ, 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 ΛΔ 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, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps.