Socially Stable Matchings

TL;DR

This paper introduces the concept of socially stable matchings in two-sided markets, accounting for limited communication, and provides complexity results and an approximation algorithm for finding maximum socially stable matchings.

Contribution

It formalizes socially stable matchings considering social communication constraints and proves NP-hardness and approximation bounds for the problem.

Findings

Maximum socially stable matching is NP-hard.

Approximation algorithm achieves a 3/2-approximation ratio.

Hardness of approximation within 3/2-{ extepsilon} under the unique games conjecture.

Abstract

In two-sided matching markets, the agents are partitioned into two sets. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. In this paper we study a variant of stable matching motivated by the fact that, in most centralized markets, many agents do not have direct communication with each other. Hence even if some blocking pairs exist, the agents involved in those pairs may not be able to coordinate a deviation. We model communication channels with a bipartite graph between the two sets of agents which we call the social graph, and we study socially stable matchings. A matching is socially stable if there are no blocking pairs that are connected by an edge in the social graph. Socially stable matchings vary in size and so we look for a maximum socially stable matching. We prove that this problem is NP-hard and, assuming the unique games conjecture, hard to approximate within a factor of 3/2-{\epsilon}, for any constant {\epsilon}>0. We complement the hardness results with a 3/2-approximation algorithm.