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. The classical notion of optimality in this setting is
stability. However stable matchings, and more generally, popular matchings need
not exist when G is non-bipartite. Unlike popular matchings, Copeland winners
always exist 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 P=NP.
We introduce a relaxation of both popular matchings and Copeland winners
called weak Copeland winners. These are matchings with Copeland score at least
μ/2, where μ is the total number of matchings in G; the maximum
possible Copeland score is (μ−1/2). We show a fully polynomial-time
randomized approximation scheme to compute a matching with Copeland score at
least μ/2⋅(1−ε) for any ε>0