Capacity Variation in the Many-to-one Stable Matching

TL;DR

This paper investigates the computational complexity of optimizing hospital capacity variations in many-to-one stable matching problems to improve residents' outcomes, revealing NP-completeness and inapproximability results.

Contribution

It establishes the NP-completeness and inapproximability of capacity variation problems in stable matching, including both expansion and reduction scenarios, under various preference assumptions.

Findings

Optimal capacity expansion is NP-complete.

Capacity reduction problems are also NP-complete.

Maximizing matching size under incomplete preferences is studied.

Abstract

The many-to-one stable matching problem provides the fundamental abstraction of several real-world matching markets such as school choice and hospital-resident allocation. The agents on both sides are often referred to as residents and hospitals. The classical setup assumes that the agents rank the opposite side and that the capacities of the hospitals are fixed. It is known that increasing the capacity of a single hospital improves the residents' final allocation. On the other hand, reducing the capacity of a single hospital deteriorates the residents' allocation. In this work, we study the computational complexity of finding the optimal variation of hospitals' capacities that leads to the best outcome for the residents, subject to stability and a capacity variation constraint. First, we show that the decision problem of finding the optimal capacity expansion is NP-complete and the corresponding optimization problem is inapproximable within a certain factor. This result holds under strict and complete preferences, and even if we allocate extra capacities to disjoint sets of hospitals. Second, we obtain analogous computational complexity results for the problem of capacity reduction. Finally, we study the variants of these problems when the goal is to maximize the size of the final matching under incomplete preference lists.