Optimal Capacity Modification for Stable Matchings with Ties

TL;DR

This paper studies capacity augmentation in the Hospitals/Residents problem with ties, focusing on strong stability, which may fail to exist in general. It introduces MinSum and MinMax augmentation objectives under both strict and tied preferences, establishing a polynomial-time algorithm for MinSum-SS and NP-hardness/inapproximability results for MinSum-COST and MinMax-SS, with a polynomial-time solution when hospital ties are bounded by $\ell+1$ yielding resident-optimal outcomes. The authors extend the MinSum framework to forced edges via a pruned instance and prove an analog of the Rural Hospitals theorem for optimal augmentations. They provide hardness reductions from SAT variants to demonstrate the computational limits of these augmentation problems and also study the bounded-tie case to identify tractable regimes and resident-optimal guarantees. Overall, the work advances capacity-design methodologies for stable matchings with ties, offering efficient algorithms and delineating clear hardness boundaries with implications for centralized matching systems.

Abstract

We consider the Hospitals/Residents (HR) problem in the presence of ties in preference lists. Among the three notions of stability, viz. weak, strong, and super stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and practically; 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. First, we show that if ties are allowed in residents' preference lists, it may not be possible to augment the hospital quotas to obtain an instance that admits a strongly stable matching. When residents' preference lists are strict, 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, whereas the MINMAX problem is NP-hard. We prove an analogue of the Rural Hospitals theorem for the MINSUM problem. When each hospital incurs a cost for a unit increase in its quota, the MINSUM problem becomes NP-hard, even for 0/1 costs. In fact, we show that the problem cannot be approximated to any multiplicative factor. We also present a polynomial-time algorithm for optimal MINSUM augmentation when a specified subset of edges is required to be included in the matching. We show that the MINMAX problem is NP-hard in general. 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 $\ell$. Amongst all instances obtained by at most $\ell$ augmentations per hospital, our algorithm produces a strongly stable matching that is best for residents.

Figures (4)

• Figure 1: Gadget $G_s$ used in the hardness reduction for \ref{['prob:minsumcost']} problem (i) Preference lists of residents in the gadget $G_{s}$. (ii) Preference list of the hospital in the gadget $G_{s}$.
• Figure 2: Preferences of global vertices in the hardness reduction for \ref{['prob:minsumcost']} problem. We assume that the variables $X_i,X_j$ and $X_k$ appear in three clauses. (i) Preference lists of the residents $a_i,a_j$ and $a_k$. (ii) Preference list of global hospitals $v_i, v_j$ and $v_k$.
• Figure 3: (i) Preference lists of residents in the gadget $G_{s}$. (ii) Preference lists of hospitals in the gadget $G_{s}$.
• Figure 6: An example instance $G$ for showing the non monotonicity property of strongly stable matching problem.

Theorems & Definitions (67)

• Definition 1.1.1: Strong stability:
• Theorem 1.1.2
• Theorem 1.1.3
• Theorem 1.1.4
• Theorem 1.1.5
• Theorem 1.1.6
• Lemma 1.2.1
• proof
• Claim 1.2.2
• proof