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Theoretical improvements in algorithmic efficiency for network flow problems (1972)
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BibTeX
@MISC{Edmonds72theoreticalimprovements,
author = {Jack Edmonds and Richard M. Karp},
title = {Theoretical improvements in algorithmic efficiency for network flow problems},
year = {1972}
}
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Abstract
This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on ... the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out that an improper choice of flow augmenting paths can lead to severe computational difficulties. Then rules of choice that avoid these difficulties are given. We show that, if each flow augmentation is made along an augmenting path having a minimum number of arcs, then a maximum flow in an n-node network will be obtained after no more than ~(n a- n) augmentations; and then we show that if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacities are integral, a maximum flow will be determined within at most 1 + logM/(M--1) if(t, S) augmentations, wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut. Next a new algorithm is given for the minimum-cost flow problem, in which all shortest-path computations are performed on networks with all weights nonnegative. In particular, this
Keyphrases
theoretical improvement network flow problem maximum flow algorithmic efficiency maximum flow problem new algorithm general minimum-cost flow problem n-node network ford-fulkerson labeling method shortest-path computation minimum number upper bound flow augmentation severe computational difficulty maximum number improper choice hitchcock transportation problem minimum-cost flow problem flow change maximum increase flow value
