Abstract

We show that a maximum flow in a network with n vertices can be computed deterministically in O(n³/log n) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n³). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(nS/3(log n)4/3), in contrast with f~(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(nal2rn 1/2 (log n) 312 + n 2 (log n) 2) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n3/2ml/2 + n2{log U) ~/2) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms.

Citations