Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond

TL;DR

This paper presents a polynomial-delay, polynomial-space algorithm for enumerating all large maximal common independent sets in two matroids, extending to matroid matchings and applications like small minimal connected vertex covers.

Contribution

It introduces a novel enumeration algorithm for large maximal common independent sets in two matroids with cardinality constraints, and extends the approach to matroid matchings and related problems.

Findings

Polynomial delay and space complexity for enumeration.

Extension to matroid matching and related combinatorial problems.

Application to enumerating small minimal connected vertex covers.

Abstract

Finding a maximum cardinality common independent set in two matroids (also known as \textsc{Matroid Intersection}) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an ``intersection'' of these problems: Given two matroids and a threshold $\tau$, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least $\tau$. We show that this problem can be solved in polynomial delay and polynomial space. Moreover, our technique can be extended to a more general problem, which is relevant to Matroid Matching. We give a polynomial-delay and polynomial-space algorithm for enumerating all maximal ``matchings'' with cardinality at least $\tau$, assuming that the optimization counterpart is ``tractable'' in a certain sense. This extension allows us to enumerate small minimal connected vertex covers in subcubic graphs. We also discuss a framework to convert enumeration with cardinality constraints into ranked enumeration.