Excluding Single-Crossing Matching Minors in Bipartite Graphs

TL;DR

This paper generalizes the class of bipartite graphs for which the permanent of associated matrices can be computed efficiently, linking graph minor exclusion to computational complexity of counting perfect matchings.

Contribution

It introduces a broad class of bipartite graphs extending previous results, unifies structural graph theory with computational complexity, and establishes hardness results for certain minors.

Findings

Exclusion of certain matching minors allows efficient permanent computation.

Identifies a new class of bipartite graphs generalizing planar and $K_{3,3}$-free graphs.

Proves $ ext{ extbf{ extit{ ext{#P}}}}$-hardness for graphs excluding $K_{5,5}$ as a matching minor.

Abstract

\noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $\#\mathsf{P}$-hard. In 1913 P\'olya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a \{matching minor}. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond $K_{3,3}.$ Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the {permanent} of the corresponding $(0,1)$-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing-minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains $\#\mathsf{P}$-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes.