An efficient implementation of a scaling minimum-cost flow algorithm

The scaling push-relabel method is an important theoretical development in the area of minimum-cost ow algorithms. We study practical implementations of this method. We are especially interested in heuristics which improve real-life performance of the method. Our implementation works very well over a wide range of problem classes. In our experiments, it was always competitive with the established codes, and usually outper-formed these codes by a wide margin. Some heuristics we develop may apply to other network algorithms. Our experimental work on the minimum-cost ow problem motivated theoretical work on related problems. of Technology Licensing, and a grant form the Powell Foundation. 0 1 1. Introduction. Signiicant theoretical progress has been made recently in the area of minimum-cost ow algorithms (see 1, 18]). Practical performance evaluation of some of these algorithms is just starting 23]. Detailed studies of somewhat older methods include investigations of network simplex 21], cost-scaling 8], and relaxation 5]. In this paper we continue our work 17] on implementing one of the recent methods, the successive approximation push-relabel method of Goldberg and Tarjan 15, 20]. This method combines and extends the ideas of cost-scaling of RR ock 30] (see also 8]), the push-relabel maximum ow method of Goldberg and Tarjan 14, 19], and the relaxation method of Bertsekas 4]. This new method looks promising for two reasons. First, the inner loop of the method is based on the push-relabel algorithm for the maximum ow problem, which in that context has been shown superior to previous codes in several experimental studies 2, 3, 9, 11, 22, 28]. Second, the successive approximation technique used in the method requires fewer iterations of the inner loop compared to the closely related cost-scaling. Performance of the previous implementations of the method 7, 17] is mixed: on some problem classes these implementations work well, while on other classes { not so well. The implementation described in this paper works better then our previous code SPUR 17] on most problems; in particular it does quite well on the classes where SPUR performed relatively poorly. The improvement is due to heuristics that improve the practical performance of the method (but not its theoretical worst-case bound). Many ideas for such heuristics were proposed in 7, 17, 20] and some have been shown to be eeective. Our current implementation succeeds in using two additional heuristics to update prices during the computation. These heuristics are closely related to …