Abstract. All previously known efftcient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the preflow concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense. graphs, achieving an O(n)) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in O(nm log(n’/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efticient distributed and parallel implementations. A parallel implementation running in O(n’log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach-Vishkin algorithm, which also uses n processors but requires O(n’) space.

Abstract not found

In this paper, we present improved polynomial time algorithms for the max flow problem defined on a network with n nodes and m arcs. We show how to solve the max flow problem in O(nm) time, improving upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm log m/(n log n) n) time. In the case that m = O(n), we improve the running time to O(n 2 / log n). We further improve the running time in the case that U ∗ = Umax/Umin is not too large, where Umax denotes the largest finite capacity and Umin denotes the smallest non-zero capacity. If log(U ∗ ) = O(n 1/3 log −3 n), we show how to solve the max flow problem in O(nm / log n) steps. In the case that log(U ∗ ) = O(log k n) for some fixed positive integer k, we show how to solve the max flow problem in Õ(n8/3) time. This latter algorithm relies on a subroutine for fast matrix multiplication. 1