Abstract
We present quantum algorithms for the following matching problems in unweighted and weighted graphs with n vertices and m edges: Finding a maximal matching in general graphs in time $O(\sqrt{nm}\log^{2}n)$. Finding a maximum matching in general graphs in time $O(n\sqrt{m}\log^{2}n)$. • Finding a maximum weight matching in bipartite graphs in time $O(n\sqrt{m}N\log^{2}n)$, where N is the largest edge weight. Our quantum algorithms are faster than the best known classical deterministic algorithms for the corresponding problems. In particular, the second result solves an open question stated in a paper by Ambainis and Spalek (Proceedings of STACS’06, pp. 172–183, 2006).
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