Finding Minimum-Cost Circulations by Successive Approximation

We develop a new approach to solving minimum-cost circulation problems. Our approach combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling. We measure the accuracy of a solution by the amount that the complementary slackness conditions are violated. We propose a simple minimum-cost circulation algorithm, one version of which runs in O ( n 3 log( nC )) time on an n -vertex network with integer arc costs of absolute value at most C . By incorporating sophisticated data structures into the algorithm, we obtain a time bound of O ( nm log( n 2 / m )log( nC )) on a network with m arcs. A slightly different use of our approach shows that a minimum-cost circulation can be computed by solving a sequence of O ( n log( nC )) blocking flow problems. A corollary of this result is an O ( n 2 (log n )log( nC ))-time, m -processor parallel minimum-cost circulation algorithm. Our approach also yields strongly polynomial minimum-cost circulation algorithms. Our results provide evidence that the minimum-cost circulation problem is not much harder than the maximum flow problem. We believe that a suitable implementation of our method will perform extremely well in practice.