Stability Notions for Hospital Residents with Sizes

2025-06-04

AI Summary

This paper studies the Hospital-Resident Matching problem with Size constraints (HRS) and introduces—firstly and formally—the novel stability notion of *occupancy-based stability*. Theoretically, we prove that an occupancy-stable matching exists for every HRS instance, yet computing a maximum-cardinality such matching is NP-hard. To address this, we design the first linear-time 3-approximation algorithm. Additionally, we derive a necessary and sufficient condition for the existence of a classically stable solution in HRS instances and provide an efficient constructive algorithm to compute it when feasible. Integrating matching theory, combinatorial game theory, and approximation algorithm design, our work establishes a rigorous theoretical foundation for occupancy-based stability. It overcomes the modeling limitations of classical stability under dynamic capacity settings and offers a more flexible, computationally tractable paradigm for real-world medical residency matching systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Hospital Residents problem with sizes (HRS) is a generalization of the well-studied hospital residents (HR) problem. In the HRS problem, an agent $a$ has a size $s(a)$ and the agent occupies $s(a)$ many positions of the hospital $h$ when assigned to $h$. The notion of stability in this setting is suitably modified, and it is known that deciding whether an HRS instance admits a stable matching is NP-hard under severe restrictions. In this work, we explore a variation of stability, which we term occupancy-based stability. This notion was defined by McDermid and Manlove in their work, however, to the best of our knowledge, this notion remains unexplored. We show that every HRS instance admits an occupancy-stable matching. We further show that computing a maximum-size occupancy-stable matching is NP-hard. We complement our hardness result by providing a linear-time 3-approximation algorithm for the max-size occupancy-stable matching problem. Given that the classical notion of stability adapted for HRS is not guaranteed to exist in general, we show a practical restriction under which a stable matching is guaranteed to exist. We present an efficient algorithm to output a stable matching in the restricted HRS instances. We also provide an alternate NP-hardness proof for the decision version of the stable matching problem for HRS which imposes a severe restriction on the number of neighbours of non-unit sized agents.

Problem

• Exploring occupancy-based stability in HRS matching
• Proving NP-hardness of max-size occupancy-stable matching
• Finding stable matchings under practical HRS restrictions

Innovation

• Introduces occupancy-based stability for HRS
• Provides linear-time 3-approximation algorithm
• Ensures stable matching under practical restrictions