Approximate max flow on small depth networks

The author considers the maximum flow problem on directed acyclic networks with m edges and depth r (length of the longest s-t path). The main result is a new deterministic algorithm for solving the relaxed problem of computing an s-t flow of value at least (1-ε) of the maximum flow. For instances where r and ε^-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, he obtains an NC O(m) processor algorithm to find a bipartite matching of cardinality (1-ε) of the maximum (for ε^-1 = O(polylog(m))). The parallel bounds are based on a novel approach to the blocking flow problem that produces fractional valued flow augmentations even when capacities are integral. She shows that a fractional flow on any network with integral capacities can be rounded in polylogarithmic time to an integral flow of no smaller value using O(m) processors. Hence, within the same resource bounds, an integral flow can be obtained when desired