Optimal Capacity Modification for Stable Matchings with Ties

TL;DR

This paper investigates how to optimally increase hospital capacities in the Hospitals/Residents problem with ties to guarantee the existence of strongly stable matchings, providing algorithms and complexity results for different optimization criteria.

Contribution

It introduces polynomial-time algorithms for MINSUM capacity augmentation, establishes NP-hardness for the cost and MINMAX variants, and adapts the rural hospitals theorem to this setting.

Findings

MINSUM augmentation can be solved in polynomial time.

Cost-aware MINSUM augmentation is NP-hard and inapproximable.

For ties of length at most ℓ+1, a polynomial-time algorithm guarantees a strongly stable matching with minimal capacity increase.

Abstract

We consider the Hospitals/Residents (HR) problem in the presence of ties in hospital preferences. Among the three notions of stability, namely weak stability, strong stability, and super-stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and in practice; however, its existence is not guaranteed. In this paper, our objective is to optimally increase the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. We explore two natural optimization criteria: (i) minimizing the total capacity increase across all hospitals (MINSUM) and (ii) minimizing the maximum capacity increase for any hospital (MINMAX). We show that the MINSUM problem admits a polynomial-time algorithm. We also establish an analog of the well-known rural hospitals theorem [Gale & Sotomayor, 1985; Roth, 1986], adapted to the MINSUM augmentation setting. We consider a generalization of the MINSUM problem in which each hospital incurs a cost per unit increase in its quota. We show that the cost version of the MINSUM problem is NP-hard and inapproximable within any multiplicative factor, even if the costs are zero or one. For the MINSUM objective with a set of forced edges, we give a polynomial-time algorithm. In contrast to the above results for the MINSUM problem, we show that the MINMAX problem is NP-hard. When hospital preference lists have ties of length at most $\ell+1$, we give a polynomial-time algorithm that increases each hospital's quota by at most $\ell$, ensuring the resulting instance admits a strongly stable matching. Moreover, among all such augmentations, our algorithm outputs the best strongly stable matching from the residents' perspective.