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

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.

12 pages

References in corpus (1)

• Counting perfect matchings in graphs that exclude a single-crossing minor