SIAM Journal on Computing, Ahead of Print.
We study the complexity of approximating the value of the independent set polynomial $Z_G(\lambda)$ of a graph $G$ with maximum degree $\Delta$ when the activity $\lambda$ is a complex number. When $\lambda$ is real, the complexity picture is well understood, and is captured by two real-valued thresholds $\lambda^*$ and $\lambda_c$, which depend on $\Delta$ and satisfy $0<\lambda^*<\lambda_c$. It is known that if $\lambda$ is a real number in the interval $(-\lambda^*,\lambda_c)$ then there is a fully polynomial time approximation scheme (FPTAS) for approximating $Z_G(\lambda)$ on graphs $G$ with maximum degree at most $\Delta$. On the other hand, if $\lambda$ 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 $\lambda^*$ and $\lambda_c$ on the $\Delta$-regular tree. The “occupation ratio” of a $\Delta$-regular tree $T$ is the contribution to $Z_T(\lambda)$ from independent sets containing the root of the tree, divided by $Z_T(\lambda)$ itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if $\lambda\in [-\lambda^*,\lambda_c]$. Unsurprisingly, the case where $\lambda$ is complex is more challenging. It is known that there is an FPTAS when $\lambda$ is a complex number with norm at most $\lambda^*$ and also when $\lambda$ is in a small strip surrounding the real interval $[0,\lambda_c)$. 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 $\lambda$ for which the occupation ratio of the $\Delta$-regular tree converges. These values carve a cardioid-shaped region $\Lambda_\Delta$ in the complex plane, whose boundary includes the critical points $-\lambda^*$ and $\lambda_c$. 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. Further, on the negative real axis, when $\lambda < - \lambda^*$, 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 Vondrák. Our proof techniques are based around tools from complex analysis---specifically the study of iterative multivariate rational maps.

中文翻译：

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复平面上独立集多项式的不逼近

《 SIAM计算杂志》，预印本。
当活动$ \ lambda $是复数时，我们研究以最大度数\\ Delta $逼近图$ G $的独立集合多项式$ Z_G（\ lambda）$的值的复杂性。当$ \ lambda $是实数时，复杂度图是很好理解的，并由两个实值阈值$ \ lambda ^ * $和$ \ lambda_c $捕获，这两个阈值取决于$ \ Delta $并满足$ 0 <\ lambda ^ * <\ lambda_c $。已知如果$ \ lambda $是区间$（-\ lambda ^ *，\ lambda_c）$中的实数，则存在一个完全多项式时间近似方案（FPTAS），用于近似$ Z_G（\ lambda）$以最大度数最多$ \ Delta $绘制$ G $。另一方面，如果$ \ lambda $是（闭合）区间之外的实数，则近似值为NP-hard。建立此图的关键是解释规则树\\ Delta $上的阈值$ \ lambda ^ * $和$ \ lambda_c $。$ \ Delta $规则树$ T $的“占有率”是包含树根的独立集合对$ Z_T（\ lambda）$的贡献，除以$ Z_T（\ lambda）$本身。当且仅当$ \ lambda \ in [-\ lambda ^ *，\ lambda_c] $中时，该占用率随着树的高度增长而收敛到一个极限。毫不奇怪，$ \ lambda $复杂的情况更具挑战性。众所周知，当$ \ lambda $是一个复数且范数最多为$ \ lambda ^ * $时，并且$ \ lambda $处于围绕实区间$ [0，\ lambda_c）的小条中时，存在FPTAS $。但是，这些结果都不能完全捕捉到何时可能近似的事实。Peters and Regts确定了$ \ lambda $的复数值，$ \ Delta $-常规树的占有率收敛。这些值在复平面上刻出一个心形形状的区域$ \ Lambda_ \ Delta $，其边界包括临界点$-\ lambda ^ * $和$ \ lambda_c $。在实际情况下，受图片的激励，他们询问$ \ Lambda_ \ Delta $是否标记为一般复数值$ \ lambda $的真实近似阈值。我们的主要结果表明，对于$ \ Lambda_ \ Delta $之外的每个$ \ lambda $，在最大度数为$ \ Delta $的图形$ G $上逼近$ Z_G（\ lambda）$的问题确实是NP-hard 。实际上，当$ \ lambda $在$ \ Lambda_ \ Delta $之外并且不是正实数时，我们得出的更强的结果是，逼近$ Z_G（\ lambda）$实际上是\＃P-hard。此外，在负实轴上，当$ \ lambda <-\ lambda ^ * $时，我们证明甚至很难确定$ Z_G（\ lambda）> 0 $是否很难解决哈维，斯里瓦斯塔瓦和冯德拉克的猜想。我们的证明技术基于复杂分析的工具-特别是对迭代多元有理图的研究。