Popular Matchings: Structure and Cheating Strategies

Abstract

We consider the cheating strategies for the popular matchings problem.
Let G = (\A \cup \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 a_1 be the sole manipulative agent who is aware of the true
preference lists of all other agents.
The goal of a_1 is to falsify her preference list to get better always, that is, to improve the set of posts she gets matched to in the falsified instance.
We show that the optimal cheating strategy for a single agent to get better always can be computed 
in O(m+n) time when preference lists are all strict and in O(\sqrt{n}m) time when preference lists are allowed to contain ties.
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  was open for the case of ties. We show an O(\sqrt{n}m) time algorithm to compute the set of popular pairs using the switching graph.
These results are of independent interest and answer a part of the open questions posed by McDermid and Irving.