We show that for any non-real algebraic number $q$ such that $|q-1|>1$ or $\Re(q)>\frac{3}{2}$ it is \textsc{\#P}-hard to compute a multiplicative (resp. additive) approximation to the absolute value (resp. argument) of the chromatic polynomial evaluated at $q$ on planar graphs. This implies \textsc{\#P}-hardness for all non-real algebraic $q$ on the family of all graphs. We moreover prove several hardness results for $q$ such that $|q-1|\leq 1$. Our hardness results are obtained by showing that a polynomial time algorithm for approximately computing the chromatic polynomial of a planar graph at non-real algebraic $q$ (satisfying some properties) leads to a polynomial time algorithm for \emph{exactly} computing it, which is known to be hard by a result of Vertigan. Many of our results extend in fact to the more general partition function of the random cluster model, a well known reparametrization of the Tutte polynomial.

翻译：我们证明，对于任意满足 $|q-1|>1$ 或 $\Re(q)>\frac{3}{2}$ 的非实代数数 $q$，在平面图上计算色多项式在 $q$ 处取值的绝对值（或辐角）的乘法（或加法）逼近是 \textsc{\#P}-困难的。这表明在所有图族上，对于所有非实代数数 $q$ 均为 \textsc{\#P}-困难。此外，我们对满足 $|q-1|\leq 1$ 的 $q$ 证明了若干困难性结果。我们的困难结果通过如下方式获得：表明若存在多项式时间算法在非实代数数 $q$（满足特定性质）下近似计算平面图的色多项式，则可导出其精确计算的多项式时间算法——而由 Vertigan 的结果已知精确计算是困难的。实际上，我们的许多结论可推广至随机簇模型的更一般配分函数，即 Tutte 多项式的著名重新参数化形式。