Parameterizing the Permanent: Hardness for fixed excluded minors

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 as a minor, 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.