Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The 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 \(M'\): here each vertex casts a vote for the matching in \(\{M,M'\}\) in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes \(\texttt {NP}\)-complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and \(\texttt {NP}\)-complete when n has the other parity.

我们的输入是n个顶点上的完整图G，其中每个顶点对G中的所有其他顶点都有严格的排名。目的是在流行的G中构造一个匹配项。如果M不输给任何匹配的\（M'\），则匹配的M很流行：在此，每个顶点都会为\（\ {M，M'\} \）中的匹配投票它得到了更好的分配。流行匹配不必在给定实例G中存在，流行匹配问题是确定是否存在。G中的流行匹配问题很容易解决 n。令人惊讶的是，对于n，问题变成\（\ texttt {NP} \）- complete，如此处所示。这是为数不多的图论的问题之一有效地解时Ñ具有一个奇偶校验和\（\ texttt {NP} \） -complete时Ñ的另一奇偶校验。