TY - JOUR

T1 - Minimum-Cost Flows in Unit-Capacity Networks

AU - Goldberg, Andrew V.

AU - Hed, Sagi

AU - Kaplan, Haim

AU - Tarjan, Robert E.

N1 - Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - We consider combinatorial algorithms for the minimum-cost flow problem on networks with unit capacities, and special cases of the problem. Historically, researchers have developed special-purpose algorithms that exploit unit capacities. In contrast, for the maximum flow problem, the classical blocking flow and push-relabel algorithms for the general case also have the best bounds known for the special case of unit capacities. In this paper we show that the classical blocking flow push-relabel cost-scaling algorithms of Goldberg and Tarjan (Math. Oper. Res. 15, 430–466, 1990) for general minimum-cost flow problems achieve the best known bounds for unit-capacity problems as well. We also develop a cycle-canceling algorithm that extends Goldberg’s shortest path algorithm (Goldberg SIAM J. Comput. 24, 494–504, 1995) to minimum-cost, unit-capacity flow problems. Finally, we combine our ideas to obtain an algorithm that solves the minimum-cost bipartite matching problem in O(r1 / 2mlog C) time, where m is the number of edges, C is the largest arc cost (assumed to be greater than 1), and r is the number of vertices on the small side of the vertex bipartition. This result generalizes (and simplifies) a result of Duan et al. (2011) and solves an open problem of Ramshaw and Tarjan (2012).

AB - We consider combinatorial algorithms for the minimum-cost flow problem on networks with unit capacities, and special cases of the problem. Historically, researchers have developed special-purpose algorithms that exploit unit capacities. In contrast, for the maximum flow problem, the classical blocking flow and push-relabel algorithms for the general case also have the best bounds known for the special case of unit capacities. In this paper we show that the classical blocking flow push-relabel cost-scaling algorithms of Goldberg and Tarjan (Math. Oper. Res. 15, 430–466, 1990) for general minimum-cost flow problems achieve the best known bounds for unit-capacity problems as well. We also develop a cycle-canceling algorithm that extends Goldberg’s shortest path algorithm (Goldberg SIAM J. Comput. 24, 494–504, 1995) to minimum-cost, unit-capacity flow problems. Finally, we combine our ideas to obtain an algorithm that solves the minimum-cost bipartite matching problem in O(r1 / 2mlog C) time, where m is the number of edges, C is the largest arc cost (assumed to be greater than 1), and r is the number of vertices on the small side of the vertex bipartition. This result generalizes (and simplifies) a result of Duan et al. (2011) and solves an open problem of Ramshaw and Tarjan (2012).

KW - Algorithms

KW - Assignment problem

KW - Bipartite matching

KW - Minimum-cost flows

UR - http://www.scopus.com/inward/record.url?scp=85019230683&partnerID=8YFLogxK

U2 - 10.1007/s00224-017-9776-7

DO - 10.1007/s00224-017-9776-7

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85019230683

SN - 1432-4350

VL - 61

SP - 987

EP - 1010

JO - Theory of Computing Systems

JF - Theory of Computing Systems

IS - 4

ER -