Popular Matchings in Complete Graphs

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 $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { 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 $$\texttt {NP}$$ 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 $$\texttt {NP}$$ NP -complete when n has the other parity.

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