Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914671750021120 |
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| author | Kobayashi, Yasuaki Kurita, Kazuhiro Wasa, Kunihiro |
| author_facet | Kobayashi, Yasuaki Kurita, Kazuhiro Wasa, Kunihiro |
| contents | 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 $τ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08948 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond Kobayashi, Yasuaki Kurita, Kazuhiro Wasa, Kunihiro Combinatorics Data Structures and Algorithms 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 $τ$, 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. |
| title | Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2307.08948 |