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arXiv · 2307.08948

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

Yasuaki Kobayashi·Kazuhiro Kurita·Kunihiro Wasa
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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.

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Yasuaki Kobayashi, Kazuhiro Kurita, Kunihiro Wasa. 2023-07-18. Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond. https://arxiv.org/abs/2307.08948

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