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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 deterministically using $O(n^{3/2}m^{1/2} + n^2(\log U)^{1/2} + \log U)$ flow operations and $O(\min\{nm, n^3/\log n\} + n^2(\log U)^{1/2} + \log U)$ time. We finally argue that several of our results yield parallel algorithms with optimal speedup.

An $o(n^3 )$-Time Maximum-Flow Algorithm