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

TL;DR

This paper introduces Pfaffian pairs and parities as a unified framework for efficiently counting bases in linear matroid problems, extending classical combinatorial theorems and providing polynomial-time algorithms for weighted cases.

Contribution

It generalizes Pfaffian pairs to Pfaffian parities in linear matroid problems, unifies various counting theorems, and develops derandomized and polynomial-time algorithms for counting solutions.

Findings

Unified framework for counting bases in linear matroid problems.

Derandomization of existing randomized algorithms for Pfaffian structures.

Polynomial-time algorithms for counting minimum-weight solutions.

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 examples, 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 enumerate combinatorial examples of Pfaffian pairs and parities. The variety of the examples illustrates that Pfaffian pairs and parities serve as a unified framework of efficiently countable discrete structures. Based on this framework, we derive celebrated counting theorems, such as Kirchhoff's matrix-tree theorem, Tutte's directed matrix-tree theorem, the Pfaffian matrix-tree theorem, and the Lindstr\"om-Gessel-Viennot lemma. Our study then turns to algorithmic aspects. We 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. Our algorithms make use of Frank's weight splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata-Kobayashi (2017).