Location-Restricted Stable Matching

TL;DR

This paper studies a location-restricted stable matching problem, proving NP-hardness for feasibility, analyzing special cases where solutions are polynomial, and addressing the complexity of minimizing blocking pairs.

Contribution

It introduces the LRSM problem, proves NP-hardness of finding feasible and stable matchings, and provides approximation algorithms for minimizing blocking pairs.

Findings

Feasibility checking is NP-hard.

Stable matchings may not exist even under restrictions.

Approximation algorithms are nearly tight for blocking pair minimization.

Abstract

Motivated by group-project distribution, we introduce and study stable matching under the constraint of applicants needing to share a location to be matched with the same institute, which we call the Location-Restricted Stable Matching problem (LRSM). We show that finding a feasible matching is NP-hard, making finding a feasible and stable matching automatically NP-hard. We then analyze the subproblem where all the projects have the same capacity, and the applicant population of each location is a multiple of the universal project capacity, which mimics more realistic constraints and makes finding a feasible matching in P. Even under these conditions, a stable matching (a matching without blocking pairs) may not exist, so we look for a matching that minimizes the number of blocking pairs. We find that the blocking pair minimization problem for this subproblem is inapproximable within $|A|^{1-\epsilon}$ for $|A|$ agents and provide an $|A|$-approximation algorithm to show this result is almost tight. We extend this result to show that the problem of minimizing the number of agents in blocking pairs is also inapproximable within $|A|^{1-\epsilon}$, and since there are only $|A|$ agents, this result is also almost tight.