Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs

Radu Curticapean; Mingji Xia

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arXiv · 2108.12879

Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs

Radu Curticapean·Mingji Xia

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Abstract

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.

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BibTeX RIS

Radu Curticapean, Mingji Xia. 2021-08-29. Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs. https://arxiv.org/abs/2108.12879

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