# Finding minimum-cost flows by double scaling

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## Abstract

Several researchers have recently developed new techniques that give fast algorithms for the minimum-cost flow problem. In this paper we combine several of these techniques to yield an algorithm running in O(*nm*(log log*U*) log(*nC*)) time on networks with*n* vertices,*m* edges, maximum arc capacity*U*, and maximum arc cost magnitude*C.* The major techniques used are the capacity-scaling approach of Edmonds and Karp, the excess-scaling approach of Ahuja and Orlin, the cost-scaling approach of Goldberg and Tarjan, and the dynamic tree data structure of Sleator and Tarjan. For nonsparse graphs with large maximum arc capacity, we obtain a similar but slightly better bound. We also obtain a slightly better bound for the (noncapacitated) transportation problem. In addition, we discuss a capacity-bounding approach to the minimum-cost flow problem.

## Key words

Minimum-cost flows transportation problem scaling dynamic trees minimum-cost circulations## Preview

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