Inapproximability of the independent set polynomial in the complex plane

TL;DR

This paper proves that approximating the independent set polynomial for graphs with bounded degree is NP-hard or #P-hard for complex parameters outside a specific cardioid-shaped region, extending complexity results to the complex plane.

Contribution

It establishes the complexity classification for approximating the independent set polynomial with complex activity parameters outside the known convergence region, using complex analysis techniques.

Findings

NP-hardness of approximation outside the cardioid region for complex λ

#P-hardness of approximation for non-real λ outside the region

Resolution of a conjecture on deciding positivity of Z_G(λ) for negative real λ

Abstract

We study the complexity of approximating the independent set polynomial $Z_G(\lambda)$ of a graph $G$ with maximum degree $\Delta$ when the activity $\lambda$ is a complex number. This problem is already well understood when $\lambda$ is real using connections to the $\Delta$-regular tree $T$. The key concept in that case is the "occupation ratio" of the tree $T$. This ratio is the contribution to $Z_T(\lambda)$ from independent sets containing the root of the tree, divided by $Z_T(\lambda)$ itself. If $\lambda$ is such that the occupation ratio converges to a limit, as the height of $T$ grows, then there is an FPTAS for approximating $Z_G(\lambda)$ on a graph $G$ with maximum degree $\Delta$. Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where $\lambda$ is complex is more challenging. Peters and Regts identified the complex values of $\lambda$ for which the occupation ratio of the $\Delta$-regular tree converges. These values carve a cardioid-shaped region $\Lambda_\Delta$ in the complex plane. Motivated by the picture in the real case, they asked whether $\Lambda_\Delta$ marks the true approximability threshold for general complex values $\lambda$. Our main result shows that for every $\lambda$ outside of $\Lambda_\Delta$, the problem of approximating $Z_G(\lambda)$ on graphs $G$ with maximum degree at most $\Delta$ is indeed NP-hard. In fact, when $\lambda$ is outside of $\Lambda_\Delta$ and is not a positive real number, we give the stronger result that approximating $Z_G(\lambda)$ is actually #P-hard. If $\lambda$ is a negative real number outside of $\Lambda_\Delta$, we show that it is #P-hard to even decide whether $Z_G(\lambda)>0$, 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.