Polynomial-delay enumeration of large maximal common independent sets in two matroids and beyond

Finding a maximum cardinality common independent set in two matroids (also known as 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...

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Bibliographic Details
Published in:Information and computation Vol. 304; p. 105282
Main Authors: Kobayashi, Yasuaki, Kurita, Kazuhiro, Wasa, Kunihiro
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2025
Subjects:
Matroid intersection
Matroid matching
Polynomial-delay enumeration
Ranked enumeration
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Matroid intersection
Ranked enumeration
Polynomial-delay enumeration
Matroid matching
Reverse search
ISSN:0890-5401
Online Access:Get full text
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Summary:Finding a maximum cardinality common independent set in two matroids (also known as 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 τ, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least τ. 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 τ, 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. •We give a poly-delay and space algorithm for enumerating all maximal common independent sets with cardinality at least τ.•Our algorithm can be extended into one that enumerates maximal matroid matchings under some tractability assumption.•We give a framework to convert an enumeration algorithm with cardinality constraints into a ranked enumeration algorithm.
ISSN:0890-5401
DOI:10.1016/j.ic.2025.105282