Stable Hypergraph Matching in Unimodular Hypergraphs
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| Formato: | Preprint |
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2025
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| _version_ | 1866909587712507904 |
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| author | Biró, Péter Csáji, Gergely Schlotter, Ildikó |
| author_facet | Biró, Péter Csáji, Gergely Schlotter, Ildikó |
| contents | We study the NP-hard Stable Hypergraph Matching (SHM) problem and its generalization allowing capacities, the Stable Hypergraph $b$-Matching (SH$b$M) problem, and investigate their computational properties under various structural constraints. Our study is motivated by the fact that Scarf's Lemma (Scarf, 1967) together with a result of Lovász (1972) guarantees the existence of a stable matching whenever the underlying hypergraph is normal. Furthermore, if the hypergraph is unimodular (i.e., its incidence matrix is totally unimodular), then even a stable $b$-matching is guaranteed to exist. However, no polynomial-time algorithm is known for finding a stable matching or $b$-matching in unimodular hypergraphs.
We identify subclasses of unimodular hypergraphs where SHM and SH$b$M are tractable such as laminar hypergraphs or so-called subpath hypergraphs with bounded-size hyperedges; for the latter case, even a maximum-weight stable $b$-matching can be found efficiently. We complement our algorithms by showing that optimizing over stable matchings is NP-hard even in laminar hypergraphs. As a practically important special case of SH$b$M for unimodular hypergraphs, we investigate a tripartite stable matching problem with students, schools, and companies as agents, called the University Dual Admission problem, which models real-world scenarios in higher education admissions.
Finally, we examine a superclass of subpath hypergraphs that are normal but necessarily not unimodular, namely subtree hypergraphs where hyperedges correspond to subtrees of a tree. We establish that for such hypergraphs, stable matchings can be found in polynomial time but, in the setting with capacities, finding a stable $b$-matching is NP-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable Hypergraph Matching in Unimodular Hypergraphs Biró, Péter Csáji, Gergely Schlotter, Ildikó Computer Science and Game Theory Data Structures and Algorithms We study the NP-hard Stable Hypergraph Matching (SHM) problem and its generalization allowing capacities, the Stable Hypergraph $b$-Matching (SH$b$M) problem, and investigate their computational properties under various structural constraints. Our study is motivated by the fact that Scarf's Lemma (Scarf, 1967) together with a result of Lovász (1972) guarantees the existence of a stable matching whenever the underlying hypergraph is normal. Furthermore, if the hypergraph is unimodular (i.e., its incidence matrix is totally unimodular), then even a stable $b$-matching is guaranteed to exist. However, no polynomial-time algorithm is known for finding a stable matching or $b$-matching in unimodular hypergraphs. We identify subclasses of unimodular hypergraphs where SHM and SH$b$M are tractable such as laminar hypergraphs or so-called subpath hypergraphs with bounded-size hyperedges; for the latter case, even a maximum-weight stable $b$-matching can be found efficiently. We complement our algorithms by showing that optimizing over stable matchings is NP-hard even in laminar hypergraphs. As a practically important special case of SH$b$M for unimodular hypergraphs, we investigate a tripartite stable matching problem with students, schools, and companies as agents, called the University Dual Admission problem, which models real-world scenarios in higher education admissions. Finally, we examine a superclass of subpath hypergraphs that are normal but necessarily not unimodular, namely subtree hypergraphs where hyperedges correspond to subtrees of a tree. We establish that for such hypergraphs, stable matchings can be found in polynomial time but, in the setting with capacities, finding a stable $b$-matching is NP-hard. |
| title | Stable Hypergraph Matching in Unimodular Hypergraphs |
| topic | Computer Science and Game Theory Data Structures and Algorithms |
| url | https://arxiv.org/abs/2502.08827 |