Stable Hypergraph Matching in Unimodular Hypergraphs

2025-02-12

AI Summary

This paper investigates Stable Hypergraph Matching (SHM) and its capacitated generalization, Stable Hypergraph $b$-Matching (SH$b$M)—both NP-hard problems. Addressing the absence of polynomial-time algorithms for unimodal hypergraphs, we systematically characterize their tractability boundaries: (i) maximum-weight stable $b$-matching is polynomial-time solvable on subpath hypergraphs; (ii) optimizing stable matchings remains NP-hard on laminar hypergraphs; (iii) SHM is tractable on subtree hypergraphs, whereas SH$b$M is NP-hard there. Methodologically, we integrate combinatorial optimization, total unimodularity theory, and constructive applications of Scarf’s Lemma and the Lovász Local Lemma. We further introduce a practical tripartite stable matching model—dual admission in higher education—and provide an efficient solution framework. Our results deliver a foundational solvability classification for hypergraph stability and supply novel algorithmic tools for constrained matching in structured hypergraphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 Lov'asz (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.

Problem

• Study Stable Hypergraph Matching in unimodular hypergraphs.
• Investigate tractable subclasses for SHM and SHbM.
• Explore polynomial-time solutions for specific hypergraph structures.

Innovation

• Unimodular hypergraphs stability
• Laminar subpath hypergraph algorithms
• Tripartite matching education model