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Stable Hypergraph Matching in Unimodular Hypergraphs

Péter Biró, Gergely Csáji, Ildikó Schlotter

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 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.

Stable Hypergraph Matching in Unimodular Hypergraphs

Abstract

We study the NP-hard Stable Hypergraph Matching (SHM) problem and its generalization allowing capacities, the Stable Hypergraph -Matching (SHM) 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 -matching is guaranteed to exist. However, no polynomial-time algorithm is known for finding a stable matching or -matching in unimodular hypergraphs. We identify subclasses of unimodular hypergraphs where SHM and SHM are tractable such as laminar hypergraphs or so-called subpath hypergraphs with bounded-size hyperedges; for the latter case, even a maximum-weight stable -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 SHM 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 -matching is NP-hard.

Paper Structure

This paper contains 28 sections, 23 theorems, 12 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Notation and Terminology
  6. Stable Hypergraph Matching
  7. The University Dual Admission Problem
  8. Formulating UDA as a Special Case of SHbM
  9. An XP Algorithm for UDA-MaxW-SHbM
  10. Further results.
  11. SHbM for Unimodular Hypergraphs
  12. Laminar Hypergraphs
  13. Subpath Hypergraphs
  14. SHbM for subtree hypergraphs
  15. Conclusions
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $A \in \mathbb{R}_+^{n \times m}$ be a matrix such that every column of $A$ has a nonzero element, and let $b \in \mathbb{R}^n_+$. Suppose that every row $i \in [n]$ has a strict ordering $\succ_i$ over those columns $j \in [m]$ for which $A_{ij}>0$. Then there is an extreme point of $\{x \in \m

Figures (1)

  • Figure 1: Connections between various classes of hypergraphs. An arrow from $\mathcal{A}$ to $\mathcal{B}$ means that hypergraphs with property $\mathcal{A}$ are a subset of hypergraphs with property $\mathcal{B}$. For the normality of subtree and unimodular hypergraphs, see bretto-hypergraphs and berge-hypergraphs, resp. In the figure, UDA stands for the class of hypergraphs that arise in our University Dual Admission problem, formally defined in Definition \ref{['def:uda-hypergraph']}.

Theorems & Definitions (58)

  • Lemma 2.1: Scarf scarf1967core
  • Definition 2.2
  • Theorem 2.3: Lovász
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: sec:app-proof-uda-reduces-to-shm
  • Theorem 3.4
  • proof
  • ...and 48 more