[Paper Review] Non-uniformly Stable Common Independent Sets

Kamiyama, Naoyuki|arXiv (Cornell University)|Jan 16, 2026

Game Theory and Voting Systems0 citations

TL;DR

They generalize the stable matching problem with matroid constraints and ties, and provide a polynomial-time algorithm to decide the existence of a non-uniformly stable common independent set of two matroids.

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.

Motivation & Objective

• Motivate and formalize stable matching generalizations with ties and matroid constraints.
• Define non-uniform stability as a common generalization of super-stability and strong stability.
• Develop a polynomial-time decision procedure for the existence of a non-uniformly stable common independent set.
• Show that the approach generalizes prior results on matroid-based stable matchings.

Proposed method

• Model the problem as finding a common independent set of two matroids under transitive, complete preference relations with ties.
• Introduce stability notions (weak/strong) generalized to matroids, including non-uniform stability.
• Develop subroutines to construct matroids ${\
• D\

Experimental results

Research questions

• RQ1 Can a non-uniformly stable common independent set be guaranteed to exist for given matroids and preferences?
• RQ2 What is the complexity of deciding the existence of such a set under matroid constraints?
• RQ3 How can matroid operations (contraction, deletion, direct sums) be leveraged to design a polynomial-time algorithm?
• RQ4 Do the results encompass and extend known matroid generalizations of stable matching (super-stable, strongly stable, and ties)?

Key findings

• The paper presents a polynomial-time algorithm to decide the existence of a non-uniformly stable common independent set of two matroids.
• The approach generalizes prior results on matroid-based stable matchings, including super-stable and strongly stable matchings under ties.
• The algorithm builds auxiliary matroids, uses base/pseudo-base constructions, and leverages circuit and closure properties to navigate feasible exchanges.
• Several lemmas (e.g., about circuits, closures, and stability-blocking behavior) underpin the iterative refinement process.
• A direct-sum and block-graph framework is employed to handle the interaction between the two matroids and the preference relations.