We study the complexity of approximating the independent set polynomial $Z_G(λ)$ of a graph $G$ with maximum degree $Δ$ when the activity $λ$ is a complex number. This problem is already well understood when $λ$ is real using connections to the $Δ$-regular tree $T$. The key concept in that case is the "occupation ratio" of the tree $T$. This ratio is the contribution to $Z_T(λ)$ from independent sets containing the root of the tree, divided by $Z_T(λ)$ itself. If $λ$ is such that the occupation ratio converges to a limit, as the height of $T$ grows, then there is an FPTAS for approximating $Z_G(λ)$ on a graph $G$ with maximum degree $Δ$. Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where $λ$ is complex is more challenging. Peters and Regts identified the complex 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. 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. If $λ$ is a negative real number outside of $Λ_Δ$, 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.

翻译：我们研究了当活动参数$λ$为复数时，近似计算最大度为$Δ$的图$G$的独立集多项式$Z_G(λ)$的复杂度。当$λ$为实数时，通过与$Δ$-正则树$T$的关联，该问题已得到充分理解。该情况下的核心概念是树$T$的"占据比"，即包含树根的独立集对$Z_T(λ)$的贡献与$Z_T(λ)$自身的比值。若对于某个$λ$，当$T$的高度增长时占据比收敛于极限，则对于最大度为$Δ$的图$G$，存在近似计算$Z_G(λ)$的完全多项式时间近似方案（FPTAS）；否则该近似问题是NP难的。自然地，$λ$为复数的情形更具挑战性。Peters与Regts确定了使$Δ$-正则树占据比收敛的复数值$λ$，这些值在复平面上划出一个心形区域$Λ_Δ$。受实数情形的启发，他们提出疑问：$Λ_Δ$是否标志着一般复数值$λ$的近似性真实阈值？我们的主要结果表明，对于$Λ_Δ$区域外的任意$λ$，在最大度不超过$Δ$的图$G$上近似计算$Z_G(λ)$确实是NP难的。事实上，当$λ$位于$Λ_Δ$外且非正实数时，我们给出了更强的结果：近似计算$Z_G(λ)$实际上是#P难的。若$λ$是$Λ_Δ$外的负实数，我们证明即使判定$Z_G(λ)>0$也属#P难问题，这肯定了Harvey、Srivastava与Vondrak的猜想。我们的证明技术基于复分析工具——特别是对迭代多元有理映射的研究。