Improved Time Bounds for the Maximum Flow Problem

Mathematical Programming, 1991

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on an n-vertex, m-arc network in at most nm pivots and O(nZm) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm log n). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.

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.

Networks, 2002

Let G = (N, A) be a network with a designated source node s, a designated sink node t, and a finite integral capacity ui 1 on each arc (i, j) E A. An elementary K-flow is a flow of K units from s to t such that the flow on each arc is 0 or 1. A K-route flow is a flow from s to t that may be expressed as a nonnegative linear sum of elementary K-flows. In this paper, we show how to determine a maximum K-route flow as a sequence of O(min {log (nU), K)) maximum-flow problems. This improves upon the algorithm by Kishimoto, which solves this problem as a sequence of K maximum-flow problems. In addition, we have simplified and extended some of the basic theory. We also discuss the application of our technique to Birkhoff's theorem and a scheduling problem.

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.

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.

Networks, 2009

The constrained maximum flow problem is to send the maximum possible flow from a source node s to a sink node t in a directed network subject to a budget constraint that the cost of flow is no more than D. In this paper, we consider two versions of this problem: (i) when the cost of flow on each arc is a linear function of the amount of flow; and (ii) when the cost of flow is a convex function of the amount of flow. We suggest capacity scaling algorithms that solve both versions of the constrained maximum flow problem in O((m log M) S(n, m)) time, where n is the number of nodes in the network, m is the number of arcs, M is an upper bound on the largest element in the data, and S(n, m) is the time required to solve a shortest path problem with nonnegative arc lengths. Our algorithms are modifications of the capacity scaling algorithms for the minimum cost flow and convex cost flow problems, and illustrate the power of capacity scaling algorithms to solve variants of the minimum cost flow problem in polynomial time.

International Journal of Computer Applications, 2013

In this paper a new approximation algorithm for calculating the min-cut tree of an undirected edgeweighted graph has been proposed. This algorithm runs in 2 2 (.log .) O V V V d  , where V is the number of vertices in the given graph and d is the degree of the graph. It is a significant improvement over time complexities of existing solutions. However, because of an assumption it does not produce correct result for all sort of graphs but for the dense graphs success rate is more than 90%. Moreover in the unsuccessful cases, the deviation from actual result is very less and for most of the pairs we obtain correct values of max-flow or mincut. This algorithm is implemented in JAVA language and checked for many input cases.

Computers & Operations Research, 2008

The constrained maximum flow problem is to send the maximum possible flow from a source node to a sink node in a directed capacitated network subject to a budget constraint that the total cost of the flow can be at most D. In this research, we present a double scaling algorithm whose generic version runs in O(n 2 m log m log U log(nC)) time, where n is the number of nodes in the network; m, the number of arcs; C, the largest arc cost; and U, the largest arc capacity. This running time can be further reduced to O(n 3 log m log U log(nC)) with the wave implementation of the cost scaling algorithm, and to O(nm log(n 2 /m) log m log U log(nC)) with the use of dynamic trees. These bounds are better than the current bound of O(mS(n, m, nC) log U), where S(n, m, nC) is the time to find a shortest path from a single source to all other nodes where nonnegative reduced costs are used as arc costs.

2014

The article presents the maximum parametric flow problem in discrete dynamic networks with linear capacities and zero lower bounds. Based on an approach of partitioning the values range of the parameter, the algorithm proposed for solving the problem finds the maximum flow for a sequence of the parameter values, in their increasing order. The flow is repeatedly augmented along the shortest paths from source to sink in the time-space network, avoiding the explicit time expansion of the network. In each of its iterations, besides the maximum flow, the algorithm also computes a new value of the parameter up to which the computed flow remains a maximum one. Finally, the complexity of the algorithm is computed. 2000 Mathematics Subject Classification: 90B10, 90C35, 90C47.