Semi-Popular Matchings and Copeland Winners

Given a graph $G = (V,E)$ where every vertex has a weak ranking over its neighbors, we consider the problem of computing an optimal matching as per agent preferences. Classical notions of optimality such as stability and its relaxation popularity could fail to exist when $G$ is non-bipartite. In light of the non-existence of a popular matching, we consider its relaxations that satisfy universal existence. We find a positive answer in the form of semi-popularity. A matching $M$ is semi-popular if for a majority of the matchings $N$ in $G$, $M$ does not lose a head-to-head election against $N$. We show that a semi-popular matching always exists in any graph $G$ and complement this existence result with a fully polynomial-time randomized approximation scheme (FPRAS). A special subclass of semi-popular matchings is the set of Copeland winners -- the notion of Copeland winner is classical in social choice theory and a Copeland winner always exists in any voting instance. We study the complexity of computing a matching that is a Copeland winner and show there is no polynomial-time algorithm for this problem unless $\mathsf{P} = \mathsf{NP}$.