Maximum flow algorithms and applications

Discrete Applied Mathematics, 1993

The maximum capacity augmentation algorithm is a variant of the Ford-Fulkerson labeling method in which each flow augmentation is along an augmenting path that yields the maximum increase in flow value. For networks with integer capacities this algorithm has been shown to be polynomial but not to be strongly polynomial, while for networks with real capacities it has been shown that it may require an infinite number of iterations. In this paper, we present two modified versions of the maximum capacity augmentation algorithm which we prove to be strongly polynomial even for real capacities. One of these is a simplex variant.

Mathematical Theory and Modeling, 2014

The Max-Flow Min-Cut Theorem is the most efficient result which can be used to determine the maximum value of flow by minimum value of capacities of all the cut sets in the network flows. In this paper we show that this theorem implies the some important results for bipartite graphs to obtain maximum flow in graph theory.

Journal of Advanced College of Engineering and Management

The aim of the maximum network flow problem is to push as much flow as possible between two special vertices, the source and the sink satisfying the capacity constraints. For the solution of the maximum flow problem, there exists a number of algorithms. The existing algorithms can be divided into two families. First, augmenting path algorithms that satisfy the conservation constraints at intermediate vertices and the second preflow push relabel algorithms that violates the conservation constraints at the intermediate vertices resulting incoming flow more than outgoing flow.In this paper, we study different algorithms that determine the maximum flow in the static and dynamic networks.

Discrete Applied Mathematics, 2011

Maximum-flow problems occur in a wide range of applications. Although already well-studied, they are still an area of active research. The fastest available implementations for determining maximum flows in graphs are either based on augmenting-path or on push-relabel algorithms. In this work, we present two ingredients that, appropriately used, can considerably speed up these methods. On the theoretical side, we present flow-conserving conditions under which subgraphs can be contracted to a single vertex. These rules are in the same spirit as presented by Padberg and Rinaldi (Math. Programming (47), 1990) for the minimum cut problem in graphs. These rules allow the reduction of known worst-case instances for different maximum flow algorithms to equivalent trivial instances. On the practical side, we propose a two-step max-flow algorithm for solving the problem on instances coming from physics and computer vision. In the two-step algorithm flow is first sent along augmenting paths of restricted lengths only. Starting from this flow, the problem is then solved to optimality using some known max-flow methods. By extensive experiments on instances coming from applications in theoretical physics and in computer vision, we show that a suitable combination of the proposed techniques speeds up traditionally used methods.

International Journal of Computer Applications, 2013

Today we are working with the networks all around and that's why it becomes very important to find the effective flow of the commodity within that network. This paper aims to provide an analysis of the best known algorithm for calculating maximum flow of any network and to propose an approximate algorithm, which can solve the same problem with lesser complexity, desirably lesser than the complexity of the known Stoer-Wagner algorithm. This paper addresses this problem with a new approach, which uses upper bound values of each node in the network.Results are compared with fixed number of nodes and variable number of nodes in the network. Moreover networks with variable densities are also considered. Results are obtained by programming the both algorithms in C++.Unix scripts are also used for formatting the results.

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.

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.

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.

Computing Research Repository, 2010

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a

Some of the results described in this article were presented at ICALP'06 .