A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows

We present a new approach to the minimum‐cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multivariate polynomials over a finite field of characteristic two for non‐identity with zero. In effect, we show that a minimum‐cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(klog(kn)+log²(kn)) time and using 2 ^k (kn) ^O(1) processors. Thus, in particular, for the minimum‐cost flow of value O(logn), we obtain an RNC² algorithm, improving upon time complexity of earlier NC and RNC algorithms.