Finding minimum-cost circulations by canceling negative cycles

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.