Approximate Max-Flow on Small Depth Networks

Edith Cohen

Abstract: We consider the maximum flow problem on directed acyclic networks with m edges and depthr (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 when r and $\epsilon^{-1}$ are small (i.e., $O(\operatorname{polylog}(m))$), this algorithm is in $\mathcal{N}\mathcal{C}$ and uses only $O(m)$ processors, which is a significant improvement over existing parallel algorithms. As one consequence, we obtain an $\mathcal{N}\mathcal{C }$$O(m)$ processor algorithm to find a bipartite matching of cardinality $(1- \epsilon)$ of the maximum (for $\epsilon^{-1} = O(\operatorname{polylog} (m))$). 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. For this case 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.