In the 1960s, statistical physicists discovered a fascinating algorithm for counting perfect matchings in planar graphs. Valiant later showed that the same problem is #P-hard for general graphs. Since then, the algorithm for planar graphs was extended to bounded-genus graphs, to graphs excluding $K_{3,3}$ or $K_{5}$, and more generally, to any graph class excluding a fixed minor $H$ that can be drawn in the plane with a single crossing. This stirred up hopes that counting perfect matchings might be polynomial-time solvable for graph classes excluding any fixed minor $H$. Alas, in this paper, we show #P-hardness for $K_{8}$-minor-free graphs by a simple and self-contained argument.

中文翻译：

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参数化永久：$K_8$-minor-free 图形的硬度

在 1960 年代，统计物理学家发现了一种用于计算平面图中完美匹配的迷人算法。Valiant 后来表明，同样的问题对于一般图来说是 #P-hard 问题。从那时起，平面图的算法被扩展到有界属图，不包括 $K_{3,3}$ 或 $K_{5}$ 的图，更一般地，扩展到不包括固定次要 $H$ 的任何图类可以在平面上用一个交叉点绘制。这激发了希望，对于不包括任何固定次要 $H$ 的图类，计算完美匹配可能是多项式时间可解的。唉，在本文中，我们通过一个简单且独立的论点展示了 $K_{8}$-minor-free 图的 #P-hardness。