We consider a natural generalization of the well-known problem where every vertex has a weight. We call a matching more popular than matching if the weight of vertices preferring to is larger than the weight of vertices preferring to . For this case, we show that it is -hard to find a popular matching. Our main result is a polynomial-time algorithm that delivers a popular matching or a proof for its non-existence in instances where all vertices on one side have weight for some and all vertices on the other side have weight 1.

中文翻译：

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与加权选民的热门匹配

我们考虑众所周知的问题的自然概括，即每个顶点都有一个权重。如果首选的顶点权重大于首选的顶点权重，我们称匹配比匹配更流行。对于这种情况，我们表明很难找到流行的匹配。我们的主要结果是一种多项式时间算法，在一侧的所有顶点的权重为某些而另一侧的所有顶点的权重为 1 的情况下，该算法提供流行的匹配或证明其不存在。