Excluded $t$-factors in Bipartite Graphs: Unified Framework for Nonbipartite Matchings, Restricted 2-matchings, and Matroids

TL;DR

This paper introduces a unified framework for optimal t-matchings excluding specific t-factors in bipartite graphs, generalizing several classical problems and providing algorithms and LP formulations.

Contribution

It generalizes nonbipartite matching problems to include various restrictions, offering a unified approach with min-max theorems, algorithms, and LP formulations.

Findings

Unified understanding of multiple matching problems

Development of a combinatorial algorithm for unweighted case

Linear programming formulation with dual integrality

Abstract

We propose a framework for optimal $t$-matchings excluding the prescribed $t$-factors in bipartite graphs. The proposed framework is a generalization of the nonbipartite matching problem and includes several problems, such as the triangle-free $2$-matching, square-free $2$-matching, even factor, and arborescence problems. In this paper, we demonstrate a unified understanding of these problems by commonly extending previous important results. We solve our problem under a reasonable assumption, which is sufficiently broad to include the specific problems listed above. We first present a min-max theorem and a combinatorial algorithm for the unweighted version. We then provide a linear programming formulation with dual integrality and a primal-dual algorithm for the weighted version. A key ingredient of the proposed algorithm is a technique to shrink forbidden structures, which corresponds to the techniques of shrinking odd cycles, triangles, squares, and directed cycles in Edmonds' blossom algorithm, a triangle-free $2$-matching algorithm, a square-free $2$-matching algorithm, and an arborescence algorithm, respectively.