Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Kazuki Matoya, Taihei Oki

Abstract: Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb (2004) introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy–Binet formula. This paper studies counting on linear matroid problems extending Webb’s work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. We also observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey (2009) and Cheung–Lau–Leung (2014) can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities.