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

TL;DR

This paper proves that counting perfect matchings remains #P-hard in graphs excluding a fixed minor, specifically $K_8$, challenging previous hopes of polynomial-time algorithms for such classes.

Contribution

It establishes the #P-hardness of counting perfect matchings in $K_8$-minor-free graphs, extending the understanding of computational complexity in minor-closed graph classes.

Findings

#P-hardness for $K_8$-minor-free graphs proven

Extends hardness results beyond previously known classes

Uses a simple, self-contained proof technique

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.