On multiroute maximum flows in networks

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.

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.

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.

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.

Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13, 2013

In this paper, we present improved polynomial time algorithms for the max flow problem defined on sparse networks with n nodes and m arcs. We show how to solve the max flow problem in O(nm + m 31/16 log 2 n) time. In the case that m = O(n 1.06 ), this improves upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm log m/(n log n) n) time. This establishes that the max flow problem is solvable in O(nm) time for all values of n and m. In the case that m = O(n), we improve the running time to O(n 2 / log n).

scs-europe.net

Network flow problems are among the most important ones in graph theory. Since there are many well-known polynomial algorithms for solving the classical Maximum Flow Problem, we, instead of summarising them, focus on special formulations and their transformation into the basic one, and because other graph theory problems may be formulated with the help of network flow tools, we show how to formulate the Minimum Steiner Tree Problem using the maximum network flow terminology and derive its mathematical model. Finally, we discuss the Integer Maximal Multicommodity Flow Problem. Since this network flow version belongs to the class of NP-hard combinatorial problems, for large scale instances, it must be solved by approximation or heuristic techniques. We present a stochastic heuristic approach based on a simulated annealing algorithm.

We encounter many different types of networks in our everyday lives, including electrical, telephone, cable, highway, rail, manufacturing and computer networks. Networks consist of special points called nodes and links connecting pairs of nodes called arcs. The maximum flow problem is one of the most fundamental problems in network flow theory and has been investigated extensively. From last few decades researchers have made a steady stream of innovation that have resulted in new solution methods and improvements to known methods. Continuous improvements to algorithms have been made by researchers for solving several classes of problems. In this paper recent techniques and algorithms related to parametric maximum flow problem are given.

2018

The generalized node-capacitated maximum flow problem is to send the maximum amount of flow from a source node to a sink node in a directed network with node capacities, where flow on a given arc exerts a workload proportional to its amount on its tail and head nodes; and sum of all such workloads on each node cannot exceed the node’s capacity. The problem has important applications and efficient solution methods are essential. We develop an efficient Lagrangian relaxation and branch and bound based method that solves the problem as a series of shortest path problems.

Journal of Combinatorial Optimization, 2011

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly N P-hard, even if every connected component of the conflict graph is a path of length two. In contrast to this we show that for forcing graphs the problem can be solved efficiently if fractional flow values are allowed. If on the other hand the flow values are required to be integral we provide the sharp line between polynomially solvable and strongly N P-hard instances.

Theoretical Computer Science, 2011

We introduce networks with additive losses and gains on the arcs.

In this paper, we are concerned with the maximum flow problem in the distribution network, a new kind of network recently introduced by Fang and Qi. It differs from the traditional network by the presence of the D-node through which the commodities are to be distributed proportionally. Adding D-nodes complicates the network structure. Particularly, flows in the distribution network are frequently increased through multiple cycles. To this end, we develop a type of depth-first-search algorithm which counts and finds all unsaturated subgraphs. The unsaturated subgraphs, however, could be invalid either topologically or numerically. The validity are then judged by computing the °ow increment with a method we call the multi-labeling method. Finally, we also provide a phase-one procedure for finding an initial flow.

This paper presents some modifications of Ford-Fulkerson's labeling method for solving the maximal network flow problem with application in solving the transportation and assignment problems. The modifications involve the tree representation of the nodes labeled and the edges used them. It is shown that after each flow adjustment some of the labels can be retained for the next labeling process. Through certain computational aspects it has been suggested that to indicate that with theses the primal-dual approach for solving the transportation and assignment problems is improved to certain extent.

advances engineering college journal, 2022

Discrete and continuous time dynamic flow problems have been studied for decades. The purpose of the network flow problem is to find the maximum flow that can be sent from the source node to the destination node. Our aim is to review the general class of continuous time dynamic flow problems. We discuss about static cut and generalized dynamic cut, the latter one used to prove the maximum flow minimum cut theorem in continuous case.

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.

Networks, 2014

In this article, we propose a generic decomposition scheme for the maximum concurrent flow problem. This decomposition scheme encompasses many models, including, among many others, the classical path formulation and the less studied tree formulation, where the flows of commodities sharing a same source vertex are routed on a set of trees. The pricing problem for this generic model is based on shortest-path computations. We show that the tree-based linear programming formulation can be solved much more quickly than the path or the aggregated arc-flow formulation. Some other decomposition schemes can lead to even faster resolution times. Finally, an efficient strongly polynomial-time combinatorial algorithm is proposed for the single-source case.