论文标题
近似色多项式与精确计算一样困难
Approximating the chromatic polynomial is as hard as computing it exactly
论文作者
论文摘要
我们表明，对于任何非真实代数$ q $，使$ | q-1 |> 1 $或$ \ re（q）> \ frac {3} {2} $ textsc {\ textsc {\＃p} -hard可以计算一个乘以$ promnom chn plannom conter plannom chn plannom at $ qul plannom at $ qul plannom at $ quonnome at $ quonnomial at $ qul plannom at $ qul at $ quonn ot。这意味着\ textsc {\＃p} - 所有图表家庭中所有非现实代数$ q $的hardness。此外，我们证明了$ Q $的几个硬度结果，因此$ | q-1 | \ leq 1 $。 我们的硬度结果是通过证明用于计算非现实代数$ Q $（满足某些属性）的平面图的多项式时间算法可导致\ emph {emph {确切}计算的多项式时间算法的，而这是一个很难通过Vertigan of pertigan of pertigan的计算。实际上，我们的许多结果扩展到了随机群集模型的更一般分区函数，这是Tutte多项式的众所周知的重复化。
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.
