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 ∆ 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 ∆ 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 ∆. 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 ∆-regular tree. The “occupation ratio” of a ∆-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 complex values of λ for which the occupation ratio of the ∆-regular tree converges. These values carve a cardioid-shaped region Λ∆ in the complex plane, whose boundary includes the critical points −λ∗ and λc. Motivated by the picture in the real case, they asked whether Λ∆ marks the true approximability threshold for general complex values λ. Our main result shows that for every λ outside of Λ∆, the problem of approximating ZG(λ) on graphs G with maximum degree at most ∆ is indeed NP-hard. In fact, when λ is outside of Λ∆ 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 Vondrák. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps. ∗Department of Computer Science, Rochester Institute of Technology, Rochester, NY, USA. Research supported by NSF grant CCF-1319987. †Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. ‡Department of Computer Science, University of Rochester, Rochester, NY 14627. Research supported by NSF grant CCF-0910415. ar X iv :1 71 1. 00 28 2v 2 [ cs .C C ] 1 8 Fe b 20 18