Popular Matchings -- structure and cheating strategies

TL;DR

This paper investigates the structure of popular matchings in bipartite graphs with preferences involving ties and develops algorithms to compute optimal cheating strategies for agents, addressing open questions in the field.

Contribution

It introduces a switching graph characterization for popular matchings with ties and provides efficient algorithms for agents to manipulate outcomes.

Findings

Optimal cheating strategies can be computed efficiently.

Switching graph characterization extends to ties, solving open problems.

Algorithms run in O(√n m) and O(m+n) time for different preference types.

Abstract

We consider the cheating strategies for the popular matchings problem. The popular matchings problem can be defined as follows: Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and the edges in E are ranked. Each agent ranks a subset of posts in an order of preference, possibly involving ties. A matching M is popular if there exists no matching M' such that the number of agents that prefer M' to M exceeds the number of agents that prefer M to M'. Consider a centralized market where agents submit their preferences and a central authority matches agents to posts according to the notion of popularity. Since a popular matching need not be unique, we assume that the central authority chooses an arbitrary popular matching. Let a1 be the sole manipulative agent who is aware of the true preference lists of all other agents. The goal of a1 is to falsify her preference list to get better always, that is, to improve the set of posts that she gets matched to as opposed to what she got when she was truthful. We show that the optimal cheating strategy for a single agent to get better always can be computed in O(\sqrt{n}m) time when preference lists are allowed to contain ties and in O(m+n) time when preference lists are all strict. Here n = |A| + |P| and m = |E|. To compute the cheating strategies, we develop a switching graph characterization of the popular matchings problem involving ties. The switching graph characterization was studied for the case of strict lists by McDermid and Irving (J. Comb. Optim. 2011) and it was open for the case of ties. The switching graph characterization for the case of ties is of independent interest and answers a part of the open questions posed by McDermid and Irving.