We present quantum algorithms for the following graph problems: finding a maximal bipartite matching in time O(n sqrt{m+n} log n), finding a maximal non-bipartite matching in time O(n^2 (sqrt{m/n} + log n) log n), and finding a maximal flow in an integer network in time O(min(n^{7/6} sqrt m * U^{1/3}, sqrt{n U} m) log n), where n is the number of vertices, m is the number of edges, and U <= n^{1/4} is an upper bound on the capacity of an edge.
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