Finding maximum flows in undirected graphs seems easier than bipartite matching
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximation algorithms and hardness of integral concurrent flow
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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For an undirected graph G=(V,E), the edge connectivity values between every pair of nodes of G can be succinctly recorded in a flow-equivalent tree that contains the edge connectivity value for a linear number of pairs of nodes. We generalize this result to show how we can efficiently recover a maximum set of disjoint paths between any pair of nodes of G by storing such sets for a linear number of pairs of nodes. At the heart of our result is an observation that combining two flow solutions of the same value, one between nodes s and r and the second between nodes r and t, into a feasible flow solution of value f between nodes s and t, is equivalent to solving a stable matching problem on a bipartite multigraph. Our observation, combined with an observation of Chazelle, leads to a data structure, which takes O(n^3^.^5) time to generate, that can construct the maximum number @l(u,v) of edge-disjoint paths between any pair (u,v) of nodes in time O(@a(n,n)@l(u,v)n) time.