Andrew V. Goldberg
Robert E. Tarjan
Abstract
A classical algorithm for finding a minimum-cost circulation consists of repeatedly finding a residual cycle of negative cost and canceling it by pushing enough flow around the cycle to saturate an arc. We show that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm. This gives a very simple strongly polynomial algorithm that uses no scaling. A variant of the algorithm that uses dynamic trees runs in Ο(nm(log n)min{log(nC), m log n}) time on a network of n vertices, m arcs, and arc costs of maximum absolute value C. This bound is comparable to those of the fastest previously known algorithms.
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Index Terms
- Finding minimum-cost circulations by canceling negative cycles
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Reviews
Max Garzon
Several elaborate strongly polytime algorithms are known for minimum-cost circulation on a network with flow capacities and cost-per-unit flows. The authors show here that a simple cycle-canceling greedy algorithm following the classical strategy of successive approximation leads to a strongly polytime algorithm. Although the classical strategy may run for an exponential number of iterations, the authors show how a judicious choice of residual cycle, namely the cycle with the least average cost, produces a minimum-cost circulation in O n 2m 3 log n iterations where arc costs are arbitrary real numbers, or O n 3/2m 2 m where arc values are integers; this algorithm is polynomial in the number of nodes n , the number of arcs m , and the largest absolute value of an arc cost C , where m := min log &parl0;nC&parr0; , n log &parl0;nC&parr0;, n m log n . As the authors point out, the proof, which relies crucially on duality and scaling, is not combinatorial. Another canceling-cycles strategy discovered by Weintraub and modified by Barahona and Tardos [1] also leads to strongly polytime algorithms.
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