Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems

Abstract

Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distance‐directed algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distance‐directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n2m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distance‐directed algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems.

Introducing Simplex Mass Balancing Method for Multi-Commodity Flow Network with a Separator

“…These approaches can be divided into two main branches, i.e. augmentation paths algorithms, [9,10,14,15,16,17,18,19] and pre-flow push algorithms [20,21,22,23,24,25,26,27,28]. Some novel ideas have also been discovered such as pseudo flows [29], and draining algorithm [30].…”

AO(nm log(U/n)) time maximum flow algorithm

“…The most important methods are attributed to Dinic [8], Edmonds and Karp [9], Karzanov [15], Malhotra, Kumar, and Maheshwari [16], Gabow [12], Sleator and Tarjan [18], Goldberg [13], Goldberg and Tarjan [14], Cheriyan and Maheshwari [6], Ahuja and Orlin [1,3,4], and Cheriyan, Hagerup, and Melhorn [7]. The theoretical complexities of these algorithms are shown in Table 1.…”

“…Sedeño-Noda and González-Martin (this paper) O (nm log(U/n)) unit capacity networks given by Ahuja and Orlin [4], introducing a scaling in the arc capacities. The behavior in the practice of this algorithm can be attractive, because of the good behavior in the practice of the two-phase algorithm and because it is possible to incorporate strategies to improve it.…”

A Mass Balancing Theorem for the Economical Network Flow Maximisation

“…These approaches can be divided into two main branches, i.e. augmentation paths algorithms, [9][10][11][12][13][14][15][16] and pre-flow push algorithms [17][18][19][20][21][22][23][24][25]. Some novel ideas have also been discovered such as pseudo flows [26], draining algorithm [27], postflow-pull method [28] and the mass balancing theorem (MBT) [29].…”