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 $\mathsf{P} = \mathsf{NP}$.

翻译：给定图 $G = (V,E)$，其中每个顶点对其邻居具有弱排序，我们考虑根据代理偏好计算最优匹配的问题。经典的最优性概念（如稳定性及其松弛的流行性）在 $G$ 为非二分图时可能不存在。鉴于流行匹配可能不存在，我们考虑其满足普遍存在性的松弛形式，并发现半流行性给出了肯定答案。若对于 $G$ 中的大多数匹配 $N$，匹配 $M$ 在直接比较中不输于 $N$，则称 $M$ 为半流行匹配。我们证明任意图 $G$ 中总存在半流行匹配，并给出一个全多项式时间随机近似方案作为存在性结果的补充。半流行匹配的一个特殊子类是科普兰获胜者的集合——社会选择理论中经典的科普兰获胜者概念在任意投票实例中均存在。我们研究了计算作为科普兰获胜者的匹配的复杂性，并证明除非 $\mathsf{P} = \mathsf{NP}$，该问题不存在多项式时间算法。