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The Set of Stable Matchings and the Core in a Matching Market with Ties and Matroid Constraints

Naoyuki Kamiyama

TL;DR

The paper addresses stable matchings in a many-to-one market with ties under general matroid constraints. It extends the core–stable matching correspondences known for uniform capacity constraints by leveraging matroid theory, including base–exchange and circuit concepts. The main result is that, in this broader setting, the set of weakly stable matchings $S$ is contained in the weak core $C$, the set of strongly stable matchings $SS$ coincides with the strong core $C_S$, and the set of super-stable matchings $SSS$ coincides with the super core $C_{SS}$; furthermore, $C \subseteq S$ may fail in general. Algorithmically, if independence oracles for the hospital matroids are available, one can decide non-emptiness of $SS$ and $SSS$ in polynomial time, extending the practical reach of these concepts beyond uniform matroids.

Abstract

In this paper, we consider a many-to-one matching market where ties in the preferences of agents are allowed. For this market with capacity constraints, Bonifacio, Juarez, Neme, and Oviedo proved some relationship between the set of stable matchings and the core. In this paper, we consider a matroid constraint that is a generalization of a capacity constraint. We prove that the results proved by Bonifacio, Juarez, Neme, and Oviedo can be generalized to this setting.

The Set of Stable Matchings and the Core in a Matching Market with Ties and Matroid Constraints

TL;DR

The paper addresses stable matchings in a many-to-one market with ties under general matroid constraints. It extends the core–stable matching correspondences known for uniform capacity constraints by leveraging matroid theory, including base–exchange and circuit concepts. The main result is that, in this broader setting, the set of weakly stable matchings is contained in the weak core , the set of strongly stable matchings coincides with the strong core , and the set of super-stable matchings coincides with the super core ; furthermore, may fail in general. Algorithmically, if independence oracles for the hospital matroids are available, one can decide non-emptiness of and in polynomial time, extending the practical reach of these concepts beyond uniform matroids.

Abstract

In this paper, we consider a many-to-one matching market where ties in the preferences of agents are allowed. For this market with capacity constraints, Bonifacio, Juarez, Neme, and Oviedo proved some relationship between the set of stable matchings and the core. In this paper, we consider a matroid constraint that is a generalization of a capacity constraint. We prove that the results proved by Bonifacio, Juarez, Neme, and Oviedo can be generalized to this setting.
Paper Structure (8 sections, 8 theorems, 1 equation)

This paper contains 8 sections, 8 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Setting
  5. Stable matchings
  6. Core
  7. Useful lemma
  8. Results

Key Result

Lemma 1

Let ${\bf M}$ be a matroid, and let $C_1,C_2$ be distinct circuits of ${\bf M}$ such that $C_1 \cap C_2 \neq \emptyset$. Then for every element $u \in C_1 \cap C_2$, there exists a circuit $C$ of ${\bf M}$ such that $C \subseteq (C_1 \cup C_2) - u$.

Theorems & Definitions (14)

  • Lemma 1: See, e.g., Oxley11
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4: Brualdi Brualdi69
  • Lemma 5: Gabow, Glover, and Klingman GabowGK74
  • proof
  • Theorem 1
  • proof
  • ...and 4 more