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An Exact Sublinear Algorithm for the Max-Flow, Vertex Disjoint Paths and Communication Problems on Random Graphs

This paper describes a randomized algorithm for solving the maximum-flow maximum-cut problem on connected random graphs. The algorithm is very fast—it does not look up most vertices in the graph. Another feature of this algorithm is that it almost surely provides, along with an optimal solution, a proof of optimality of the solution. In addition, the algorithm's solution is, by construction, a collection of vertex-disjoint paths which is maximum. Under a restriction on the graph's density, an optimal solution to the NP-hard communication problem is provided as well, that is, finding a maximum collection of vertex-disjoint paths between sender-receiver pairs of terminals. The algorithm lends itself to a sublogarithmic parallel and distributed implementation. Its effectiveness is demonstrated via extensive empirical study.