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Non-uniformly Stable Common Independent Sets

Naoyuki Kamiyama

TL;DR

This work generalizes stable matching with ties to a matroid setting by studying non-uniformly stable common independent sets of two matroids. It develops a polynomial-time algorithm that builds restricted matroids ${\cal M}_D\langle F\rangle$ and ${\cal M}_H\langle F\rangle$, guided by closure and circuit structure, to decide existence and construct a solution when possible. The approach unifies and extends previous matroid-based stability results and applies to many-to-many settings, underpinned by a key lemma and intricate case analyses. Overall, the method yields a constructive, efficient means to check non-uniform stability in matroid choosings, with potential applications to generalized stable matchings under matroid constraints.

Abstract

In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem of checking the existence of a common independent set satisfying non-uniform stability, which is a common generalization of super-stability and strong stability.

Non-uniformly Stable Common Independent Sets

TL;DR

This work generalizes stable matching with ties to a matroid setting by studying non-uniformly stable common independent sets of two matroids. It develops a polynomial-time algorithm that builds restricted matroids and , guided by closure and circuit structure, to decide existence and construct a solution when possible. The approach unifies and extends previous matroid-based stability results and applies to many-to-many settings, underpinned by a key lemma and intricate case analyses. Overall, the method yields a constructive, efficient means to check non-uniform stability in matroid choosings, with potential applications to generalized stable matchings under matroid constraints.

Abstract

In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem of checking the existence of a common independent set satisfying non-uniform stability, which is a common generalization of super-stability and strong stability.
Paper Structure (12 sections, 32 theorems, 21 equations, 4 algorithms)

This paper contains 12 sections, 32 theorems, 21 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics of matroids
  4. Direct sum of matroids
  5. Common independent sets
  6. Algorithm
  7. Subroutines
  8. Main algorithm
  9. Correctness
  10. Proof of Lemma \ref{['lemma:key']}
  11. Case 1
  12. Case 2

Key Result

Lemma 1

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

Theorems & Definitions (74)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1: See, e.g., Oxley11
  • Lemma 2: See, e.g., Oxley11
  • Lemma 3: See, e.g., Oxley11
  • Lemma 4: See, e.g., Oxley11
  • Lemma 5: Iri and Tomizawa IriT76
  • Lemma 6: See, e.g., IriT76
  • Lemma 7: See, e.g., Kamiyama22
  • ...and 64 more