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<λ^*<λ_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<−λ^*, we show that it is #P-hard to even decide whether Z_G(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.