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

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)

Finding a Shortest Non-zero Path in Group-Labeled Graphs

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slightly more general class of groups, Yamaguchi (2016) showed a reduction of the problem to the weighted linear matroid parity problem. In particular, some cases are solved in strongly polynomial time via the reduction with the aid of a deterministic, polynomial-time algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2017), which generalizes a well-known fact that the parity-constrained shortest path problem is solved via weighted matching. In this paper, as the first general solution independent of the group, we present a rather simple, deterministic, and strongly polynomial-time algorithm for the shortest non-zero path problem. The algorithm is based on Dijkstra's algorithm for the unconstrained shortest path problem and Edmonds' blossom shrinking technique in matching algorithms, and clarifies a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem. Furthermore, we demonstrate a faster algorithm without explicit blossom shrinking together with a dual linear programming formulation of the equivalent problem like potential maximization for the unconstrained shortest path problem.Comment: A preliminary version appeared in SODA 2020; 21 pages, 5 figure