Near-Feasible Solutions to Complex Stable Matching Problems

TL;DR

This paper shows that for complex NP-complete stable matching variants, near-feasible stable solutions always exist and can be efficiently found through an iterative rounding algorithm that minimally adjusts capacities.

Contribution

The paper introduces a polynomial-time iterative rounding method to find near-feasible stable solutions in complex stable matching problems, extending stability guarantees under capacity constraints.

Findings

Near-feasible stable solutions exist for many NP-complete variants.

An iterative rounding algorithm can efficiently find these solutions.

Small capacity modifications can restore stability in complex scenarios.

Abstract

In this paper, we demonstrate that in many NP-complete variants of the stable matching problem, such as the Stable Hypergraph Matching problem, the Stable Multicommodity Flow problem, and the College Admission problem with common quotas, a near-feasible stable solution - that is, a solution which is stable, but may slightly violate some capacities - always exists. Our results provide strong theoretical guarantees that even under complex constraints, stability can be restored with minimal capacity modifications. To achieve this, we present an iterative rounding algorithm that starts from a stable fractional solution and systematically adjusts capacities to ensure the existence of an integral stable solution. This approach leverages Scarf's algorithm to compute an initial fractional stable solution, which serves as the foundation for our rounding process. Notably, in the case of the Stable Fixtures problem, where a stable fractional matching can be computed efficiently, our method runs in polynomial time. These findings have significant practical implications for market design, college admissions, and other real-world allocation problems, where small adjustments to institutional constraints can guarantee stable and implementable outcomes.