We study efficient implementations of the push-relabel method for the maximum flow problem. The resulting codes are faster than the previous codes, and much faster on some problem families. The speedup is due to the combination of heuristics used in our implementation. We also exhibit a family of problems for which all known methods seem to have almost quadratic time growth rate.

Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worst-case time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.

We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the push-relabel method is most efficient in practice and to compare push-relabel algorithms with augmenting path algorithms. We have implemented and compared three push-relabel algorithms, three augmenting path algorithms (one of which is new), and one augment-relabel algorithm. The depth-first search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowest-level selection push-relabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.

Abstract. The maximum flow problem is a classical optimization problem with many applications. For a long time, HI-PR, an efficient implementation of the highest-label push-relabel algorithm, has been a benchmark due to its robust performance. We propose another variant of the push-relabel method, the partial augment-relabel (PAR) algorithm. Our experiments show that PAR is very robust. It outperforms HI-PR on all problem families tested, asymptotically in some cases. 1

Abstract. We describe a two-level push-relabel algorithm for the maximum flow problem and compare it to the competing codes. The algorithm generalizes a practical algorithm for bipartite flows. Experiments show that the algorithm performs well on several problem families. 1