Approximate Max-Flow on Small Depth Networks

We consider the maximum flow problem on directed acyclic networks with $m$ edges and depth $r$ (length of the longest $s$-$t$ path). Our main result is a new deterministic algorithm for solving the relaxed problem of computing an $s$-$t$ flow of value at least $(1-\epsilon)$ of the maximum flow. For instances where $r$ and $\epsilon^{-1}$ are small (i.e., $O(\polylog(m))$), this algorithm is in $\NC$ and uses only $O(m)$ processors, which is a significant improvement over existing parallel algorithms. As one consequence, we obtain an $\NC$ $O(m)$ processor algorithm to find a bipartite matching of cardinality $(1-\epsilon)$ of the maximum (for $\epsilon^{-1}= O(\polylog(m))$). We use a novel approach based on path-counts to compute blocking flows in parallel. This approach produces fractional flow even when capacities are integral. To encounter that, we provide a rounding algorithm that is of independent interest. In polylogarithmic time using $O(m)$ processors, the algorithm rounds any fractional flow on a network with integral capacities to an integral flow. The rounding technique extends to networks with costs.