MPI-I-91-120

We show that a maximum flow in a network with $n$ vertices can be computed deterministically in $O({{n^3}/{\log n}})$ time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of $O(n^3)$. The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is $O(n^{8/3}(\log n)^{4/3})$, in contrast with $\Omega(nm)$ flow operations for all previous algorithms, where $m$ denotes the number of edges in the network. A randomized version of our algorithm executes $O(n^{3/2}m^{1/2}\log n+n^2(\log n)^2/ \log(2+n(\log n)^2/m))$ flow operations with high probability. For the special case in which all capacities are integers bounded by $U$, we show that a maximum flow can be computed