Computational investigations of maximum flow algorithms

Naval Research Logistics, 1991

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(n 2 m) 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.

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 the 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 paper, we develop two new distance-directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n 2 m) time. We show that these algorithms are equivalent to Edmonds-Karp and Dinic's algorithm in the sense that they enumerate the same augmenting paths in the same sequence.

SIAM Journal on Computing, 1989

Recently, Goldberg proposed a new approach to the maximum network flow problem. The approach yields a very simple algorithm running in O(n 3) time on n-vertex networks. Incorporation of the dynamic tree data structure of Sleator and Tarjan yields a more complicated algorithm with a running time of O(nm log (n 2 /m)) on m-arc networks. Ahuja and Orlin developed a variant of Goldberg's algorithm that uses scaling and runs in O(nm + n 2 log U) time on networks with integer arc capacities bounded by U. In this paper possible improvements to the Ahuja-Orlin algorithm are explored. First, an improved running time of O(nnz + n log U/log log U) is obtained by using a nonconstant scaling factor. Second, an even better bound of O(nm + n2(log U) 1 /2) is obtained by combining the Ahuja-Orlin algorithm with the wave algorithm of Tarjan. Third, it is shown that the use of dynamic trees in the latter algorithm reduces the running time to O(nm log ((n/m)(log U)t/2 + 2)). This result shows that the combined use of three different techniques results in speed not obtained by using any of the techniques alone. The above bounds are all for a unit-cost random access machine. Also considered is a semilogarithmic computation model in which the bounds increase by an additive term of O(m log,, U), which is the time needed to read the input in the model.

Computers & Operations Research, 2004

In this paper, we generalize the capacity-scaling techniques in the design of algorithms for the maximum flow problem. Since all previous scaling max-flow algorithms use only one scale factor of value 2, we propose introducing a double capacity-scaling to imp rove and generalize them. The first capacity scaling has a variable scale factor β and the second uses the value 2. We show that, for different values of the scale factor β , both the classical scaling algorithm (with β =U) and the two-phase double scaling-capacity max-flow algorithm (with β =2) can be obtained. Moreover, theoretical complexities based on the worst-case analysis can be built depending on the values of β . In addition, a unique and simple implementation of the generalized method is possible and several strategies to improve its practical behavior can be incorporated. The paper finishes with a computational experiment that shows that the running time of capacity-scaling algorithms decreases as β increases.

Computers & Operations Research, 2012

The capacity scaling algorithm for the maximum flow problem runs in Oðnm log UÞ time where n is the number of nodes, m is the number of arcs, and U is the largest arc capacity in the network. The twophase capacity scaling algorithm reduces this bound to Oðnm logðU=nÞÞ. This running time is achieved with the restriction that flows are pushed on individual arcs while paths are being identified, but this causes slower empirical run times compared to the single-phase capacity scaling algorithm. In this research, we prove that the two-phase algorithm runs in the same theoretical time without the mentioned restriction. We also show that in practice, it runs significantly faster than the single-phase capacity scaling algorithm.

Zulfaqar J.Def. Eng. Tech, 2023

This paper aims to introduce and discuss two existing algorithms, namely Ford-Fulkerson's Algorithm and Dinic's Algorithm. These algorithms are for determining the maximum flow from source (s) to sink (t) in a flow network. A numerical example is solved to illustrate both algorithms, and to demonstrate, study, and compare the procedures at each iteration. The results show that Dinic's Algorithm returns the maximum flow that takes less number of iterations and augmentations than the Ford-Fulkerson Algorithm. In terms of complexity, the running time of Dinic's algorithm is (2), which should make it perform better on dense graphs. This goes to show that the claim by many researchers that Dinic's Algorithm is very powerful in solving big network flow problems is justified.

Lecture Notes in Computer Science

We show that a maximum flow in a network with n vertices can be computed deterministically in O(n3/logn) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n3). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(nS/3(log n)4/3), in contrast with f~(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(nal2rn 1/2 (log n) 312 + n 2 (log n) 2) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n3/2ml/2 + n2{log U) ~/2) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms.

2003

An incremental algorithm may yield an enormous computational time saving to solve a network flow problem. It updates the solution to an instance of a problem for a unit change in the input. In this paper we have proposed an efficient incremental implementation of maximum flow problem after inserting an edge in the network G. The algorithm has the time complexity of O(( n) 2 m), where n is the number of affected vertices and m is the number of edges in the network. We have also discussed the incremental algorithm for deletion of an edge in the network G. : 90B10, 90C27.

SIAM Journal on Computing, 1996

Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung und Technologie (Betreuungskennzeichen ITS 9103) gefördert. Die Verantwortung für den Inhalt dieser Veröffentlichung liegt beim Autor." An o(n 3)-Time Maximum-Flow Algorithm 1 ,2