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A Faster Algorithm for Finding the Minimum Cut in a Directed Graph
 JOURNAL OF ALGORITHMS
, 1994
"... 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 sepa ..."
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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.
New DistanceDirected Algorithms for Maximum Flow and Parametric Maximum Flow Problems
, 1987
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Distancedirected augmenting path algorithms for maximum flow and parametric maximum flow problems
 Naval Research Logistics
, 1991
"... Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distancedirected algorithms have been suggested that do not construct layered networks but instead ..."
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Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distancedirected algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distancedirected augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n 2 m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distancedirected algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems. t i
Diagnosing Infeasibilities in Network Flow Problems
 Math. Programming
, 1998
"... 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 i ..."
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Cited by 2 (0 self)
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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 NPhard. 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 preflowpush 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.
Diagnosing infeasibilities in network flow problems
, 1997
"... We consider the problem of finding a feasible flow in a directed network 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 uij. It is well known that this feasibility problem can be transformed into a maximum flow problem. It i ..."
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We consider the problem of finding a feasible flow in a directed network 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 uij. 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 in S exceeds the capacity of the arcs emanating from S. Such a set S is called a witness of infeasibility" (or, simply, a wtiitness) 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 NPhard. We also consider the problem of determining a minimal witness, 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 preflowpush algorithm to solve