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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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 105282 |
| Author | Wasa, Kunihiro Kobayashi, Yasuaki Kurita, Kazuhiro |
| Author_xml | – sequence: 1 givenname: Yasuaki surname: Kobayashi fullname: Kobayashi, Yasuaki organization: Hokkaido University, Kita 14-jo Nishi 9-choume, Kita-ku, Sapporo, 060-0814, Hokkaido, Japan – sequence: 2 givenname: Kazuhiro orcidid: 0000-0002-7638-3322 surname: Kurita fullname: Kurita, Kazuhiro email: kurita@i.nagoya-u.ac.jp organization: Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8601, Aichi, Japan – sequence: 3 givenname: Kunihiro surname: Wasa fullname: Wasa, Kunihiro organization: Hosei University, Kajinocho, Koganei, 184-8584, Tokyo, Japan |
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| Keywords | Matroid intersection Ranked enumeration Polynomial-delay enumeration Matroid matching Reverse search |
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| Title | Polynomial-delay enumeration of large maximal common independent sets in two matroids and beyond |
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