Computational investigations of maximum flow algorithms

The maximum flow algorithm is distinguished by the long line of successive contributions researchers have made in obtaining algorithms with incrementally better worst-case complexity. Some, but not all, of these theoretical improvements have produced improvements in practice. The purpose of this paper is to test some of the major algorithmic ideas developed in the recent years and to assess their utility on the empirical front. However, our study differs from previous studies in several ways. Whereas previous studies focus primarily on CPU time analysis, our analysis goes further and provides detailed insight into algorithmic behavior. It not only observes how algorithms behave but also tries to explain why algorithms behave that way. We have limited our study to the best previous maximum flow algorithms and some of the recent algorithms that are likely to be efficient in practice. Our study encompasses ten maximum flow algorithms and five classes of networks. The augmenting path algorithms tested by us include Dinic's algorithm, the shortest augmenting path algorithm, and the capacity-scaling algorithm. The preflow-push algorithms tested by us include Karzanov's algorithm, three implementations of Goldberg-Tarjan's algorithm, and three versions of Ahuja-Orlin-Tarjan's excess-scaling algorithms. Among many findings, our study concludes that the preflow-push algorithms are substantially faster than other classes of algorithms, and the highest-label preflow-push algorithm is the fastest maximum flow algorithm for which the growth rate in the computational time is O(n LS) on four out of five of our problem classes. Further, in contrast to the results of the worst-case analysis of maximum flow algorithms, our study finds that the time to perform relabel operations (or constructing the layered networks) takes at least as much computation time as that taken by augmentations and/or pushes. © 1997 Published by Elsevier Science B.V. 1. I n t r o d u c t i o n The maximum flow problem is one of the most fundamental problems in network optimization. Its intuitive appeal, mathematical simplicity, and wide applicabil i ty has made it a popular research topic * Corresponding author. 0377-2217/97/$17.00 © 1997 Published by Elsevier Science B.V. All PII S0377-2217(96)00269-X among mathematicians, operations researchers and computer scientists. The maximum flow problem arises in a wide variety of situations. It occurs directly in problems as diverse as the flow of commodit ies in pipeline networks, parallel machine scheduling, distributed computing on multi-processor computers, matrix rounding problems, the baseball el imination problem, and the statistical security of data. The maximum flow