Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our goal is to construct a matching in $G$ that is popular. A matching $M$ is popular if $M$ does not lose a head-to-head election against any matching $M'$, where each vertex casts a vote for the matching in $\{M,M'\}$ where it gets assigned a better partner. The popular matching problem is to decide whether a popular matching exists or not. The popular matching problem in $G$ is easy to solve for odd $n$. Surprisingly, the problem becomes NP-hard for even $n$, as we show here.

翻译：我们的投入是一个完整的图表$G = (V,E) $n 顶点的美元(V,E) = $n 顶点的美元(V,E) = $n 顶点的美元($美元) 。 我们的目标是用流行的美元构建匹配。 如果美元不会因为任何匹配的美元而失去头对头的选举,那么匹配美元($M = ($G) = (V,E) = $n 顶点的美元($M,$) = (美元) = (美元) 美元。 流行的匹配问题是决定流行匹配是否存在。 $G 的流行匹配问题很容易解决, 奇数美元。 奇怪的是, 如我们在这里所显示的, 问题甚至连美元都变成NP- hard。