Popular Matchings in Complete Graphs
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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.
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