In the Popular Matching problem, we are given a bipartite graph $G = (A \cup B, E)$ and for each vertex $v\in A\cup B$, strict preferences over the neighbors of $v$. Given two matchings $M$ and $M'$, matching $M$ is more popular than $M'$ if the number of vertices preferring $M$ to $M'$ is larger than the number of vertices preferring $M'$ to $M$. A matching $M$ is called popular if there is no matching $M'$ that is more popular than $M$. We consider a natural generalization of Popular Matching where every vertex has a weight. Then, we call a matching $M$ more popular than matching $M'$ if the weight of vertices preferring $M$ to $M'$ is larger than the weight of vertices preferring $M'$ to $M$. For this case, we show that it is NP-hard to find a popular matching. Our main result its a polynomial-time algorithm that delivers a popular matching or a proof for it non-existence in instances where all vertices on one side have weight $c > 3$ and all vertices on the other side have weight 1.
翻译：在大众相配问题中,我们得到一个双面图$G = (A\ cup B, E)$,对于每顶顶点$ = (A\ cup B) 美元,我们得到一个双面图$G = (A\ cup B, E)$,对于每顶点美元,我们得到对邻居的严格优惠为$V美元。如果双对匹配美元和美元,匹配美元比美元更受欢迎,那么匹配美元比美元多。如果偏爱美元对美元对美元对美元的顶点数量大于偏爱美元对美元对美元。如果双面的顶点对面点的重量比重大于每面的顶点,则匹配美元比美元更受欢迎。对于这种情况,如果每顶点偏爱美元对美元对美元对美元对美元对美元对美元。如果顶点的重量比重大于每顶点比1美元对美元,那么匹配美元是受欢迎的。我们考虑对大众匹配的自然普遍化。然后,如果每顶点对每顶点的重量比美元比美元,我们要求一个顶点对3个顶点的顶点的顶点结果。
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