Max flows in O(nm) time, or better

Lecture Notes in Computer Science, 1998

We i n troduce a gain-scaling technique for the generalized maximum ow problem. Using this technique, we present three simple and intuitive polynomial-time combinatorial algorithms for the problem. Truemper's augmenting path algorithm is one of the simplest combinatorial algorithms for the problem, but runs in exponential-time. Our rst algorithm is a polynomial-time variant o f T ruemper's algorithm. Our second algorithm is an adaption of Goldberg and Tarjan's pre owpush algorithm. It is the rst polynomial-time pre ow-push algorithm in generalized networks. Our third algorithm is a variant of the Fat-Path capacity-scaling algorithm. It is much simpler than Radzik's variant and matches the best known complexity for the problem. We discuss practical improvements in implementation.

Computers & Operations Research, 2012

The constrained maximum flow problem is a variant of the classical maximum flow problem in which the total cost of the flow from the source to sink is constrained by a budget limit. It is important to study this problem because it has many important practical applications. In this research, we present a new polynomial time algorithm that is based on the cost scaling algorithm for the minimum cost network flow problem. We prove that it runs in O(n 2 m log(nC)) worst case time.

Network Biology, 2022

In this paper, the binary representation of arc capacity has been used in developing an efficient polynomial time algorithm for the constrained maximum flow problem in directed networks. The algorithm is basically based on solving the maximum flow problem as a sequence of O(n 2) shortest path problems on residual directed networks with n nodes generated during iterations. The complexity of the algorithm is estimated to be no more than O(n 2 mr) arithmetic operations, where m denotes the number of arcs in the network, and r is the smallest integer greater than or equal to log B (B denotes the largest arc capacity in the directed network). Generalization of the algorithm has been also performed in order to solve the maximum flow problem in a directed network subject to non-negative lower bound on the flow vector. A formulation of the simple transportation problem, as a maximal network flow problem has been also performed. Numerical example has been inserted to illustrate the use of the proposed algorithm. Keywords maximum flow problem; scaling algorithm; polynomial time algorithm; augmenting path method; network flow.

1988

Goldfarb and Hao have proposed a. network simplex a,lgorithm that will solve a ma.ximum flow problem on an n-vertex, m-arc network in at most nm pivots and 0(n2m) time. In this paper we describe how to implement their algorithm to run in O(nm log n) time by using an extension of the dynamic tree data structure of Sleator and Tarjan. This bound is less than a logarithmic factor la,rger than that of any other known algorithm for the problem.

Mathematical Programming, 1990

We propose a primal network simplex algorithm for solving the maximum flow problem which chooses as the arc to enter the basis one that is closest to the source node from amongst all possible candidates. We prove that this algorithm requires at most nm pivots to solve a problem with n nodes and m arcs, and give implementations of it which run in O(nZm) time. Our algorithm is, as far as we know, the first strongly polynomial primal simplex algorithm for solving the maximum flow problem.

The Maximum Concurrent Flow Problem has shown significant utility in finding community structure in graphs. However, the time and space complexities of the formulation limit feasibility to relatively small graphs. Some of our previous work has used a heuristic to estimate the solution in larger graphs. In this paper, we use a similar method to coarsen the graph to use the full linear programming approach to solve the MCFP. Most naturally occurring graphs, such as those in social networks, follow a power law distribution of degree. Therefore, a coarsening approach that tends to group graph communities creates, with high probability, super-nodes that will not be internally separated as part of the real solution of the MCFP.

1997

Poznan, The maximum flow algorithm is distinguished by the long line of successive contributions researchers have made in ausanne, obtaining algorithms with ncrementally 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 fcus primarily on CPU time analysis, our analysis goes further and provides naterials, detailed insight into algorithmic behavior. It not only observes how algorithms behave but also tries to explain why ,,<uences quences algorithms behave that way. We have limited our study to the best previous maximum flow algorithms and some of the and gov-I> Research recent algorithms that are likely to be efficient in practice. Our study encompasses ten maximum flow algorithms and five making, classes of networks. The augmenting path algorithms tested by us include Dinic's algorithm, the shortest augmenting path made for apable of algorithm, and the capacity-scaling algorithm. The preflow-push algorithms tested by us include Karzanov's algorithm, three apable of) :-Ant implementations of Goldberg-Tarjan's algorithm, and three versions of Ahuja-Orlin-Tarjan's excess-scaling algorithms. the past Among many findings, our study concludes that the preflow-push algorithms are substantially faster than other classes of nd of this nd of this algorithms, and the highest-label preflow-push algorithm is the fastest maximum flow algorithm for which the growth rate in structions i he editors structlons the computational time is O(n t ' 5) 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.

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

Networks, 2006

We are given a directed network G = (V, A, u) with vertex set V , arc set A, a source vertex s ∈ V , a destination vertex t ∈ V , a finite capacity vector u = {u ij } ij∈A , and a positive integer m ∈ Z +. The multiroute maximum flow problem (m-MFP) generalizes the ordinary maximum flow problem by seeking a maximum flow from s to t subject to not only the regular flow conservation constraints at the vertices (except s and t) and the flow capacity constraints at the arcs, but also the extra constraints that any flow must be routed along m arc-disjoint s-t paths. In this paper, we devise two new combinatorial algorithms for m-MFP. One is based on Newton's method and another is based on augmenting-path technique. We also show how the Newton-based algorithm unifies two existing algorithms, and how the augmenting-path algorithm is strongly polynomial for case m = 2.