Algebraic Structures and Algorithms for Matching and Matroid Problems (Preliminary Version)

TL;DR

This paper introduces a new algebraic framework and algorithms for basic path-matching problems, improving efficiency for matroid intersection and non-bipartite matching by leveraging matrix multiplication techniques.

Contribution

It provides a novel algebraic characterization of basic path-matching problems and develops randomized algorithms with improved running times for key matching and matroid intersection problems.

Findings

Improved algorithm for matroid intersection with time O(nr^1.38)

Algebraic algorithm for non-bipartite matching with time O(n^w)

Resolution of a central open problem in non-bipartite matching

Abstract

Basic path-matchings, introduced by Cunningham and Geelen (FOCS 1996), are a common generalization of matroid intersection and non-bipartite matching. The main results of this paper are a new algebraic characterization of basic path-matching problems and an algorithm for constructing basic path-matchings in O(n^w) time, where n is the number of vertices and w is the exponent for matrix multiplication. Our algorithms are randomized, and our approach assumes that the given matroids are linear and can be represented over the same field. Our main results have interesting consequences for several special cases of path-matching problems. For matroid intersection, we obtain an algorithm with running time O(nr^(w-1))=O(nr^1.38), where the matroids have n elements and rank r. This improves the long-standing bound of O(nr^1.62) due to Gabow and Xu (FOCS 1989). Also, we obtain a simple, purely algebraic algorithm for non-bipartite matching with running time O(n^w). This resolves the central open problem of Mucha and Sankowski (FOCS 2004).