We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n- 2 maximum flow problems. We then show how to reduce the running time of these 2n- 2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log(n 2 /m)) for finding the minimum cut in either a directed or an undirected network. © 1994 Academic Press, Inc. 1.

Abstract not found

We consider the problem of finding a feasible flow in a directed G = (N, A) in which each node i N has a supply b(i), and each arc (i, j) A has a zero lower bound on flow and an upper bound u ij . It is well known that this feasibility problem can be transformed into a maximum flow problem. It is also well known that there is no feasible flow in the network G if and only if there is a subset S of nodes such that the net supplies of the nodes of S exceeds the capacity of the arcs emanating from S. Such a set S is called a "witness of infeasibility" (or, simply, a witness) of the network flow problem. In the case that there are many different witnesses for an infeasible problem, a small cardinality witness may be preferable in practice because it is generally easier for the user to assimilate, and may provide more guidance to the user on how to identify the cause of the infeasibility. Here we show that the problem of finding a minimum cardinality witness is NP-hard. We also consider the problem of determining a minimal witnesses, that is, a witness S such that no proper subset of S is also a witness. In this paper, we show that we can determine a minimal witness by solving a sequence of at most n maximum flow problems. Moreover, if we use the preflow-push algorithm to solve the resulting maximum flow problems and organize computations properly, then the total time taken by the algorithm is comparable to that of solving a single maximum flow problem. This approach determines a minimal cardinality witness in O(n 3 ) time using simple data structures and in O(nm log(n 2 /m)) time using the dynamic tree data structures.