Locating Sources to Meet Flow Demands in Undirected Networks

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.

European Journal of Operational Research, 2008

Given arbitrary source and target nodes s, t and an s-t-flow defined by its flow-values on each arc of a network, we consider the problem of finding a decomposition of this flow with a minimal number of s-t-paths. This problem is issued from the engineering of telecommunications networks for which the task of implementing a routing solution consists in integrating a set of end-to-end paths. We show that this problem is NP-hard in the strong sense and give some properties of an optimal solution. We then propose upper and lower bounds for the number of paths in an optimal solution. Finally we develop two heuristics based on the properties of a special set of solutions called saturating solutions.

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.

Yugoslav Journal of Operations Research, 2013

We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear single-commodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss state-of-the-art algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented.

ArXiv, 2021

This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O(F (m+ n) +mn+M) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher’s well-known method for the determination of the K best integer flows [11] at the first sub-optimal flow results in an algorithm with a running time of O(Fm(n logn+m) +M) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so-called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O(Kn3 +M). Besides, we give lower and upper bounds for the number ...

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, 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.

Algorithmica, 1992

We present two variants of the primal network simplex algorithm which solve the minimum cost network flow problem in at most O(n2m 2 log n) pivots. Here we define the network simplex method as a method which proceeds from basis tree to adjacent basis tree regardless of the change in objective function value; i.e., the objective function is allowed to increase on some iterations. The first method is an extension of the minimum mean augmenting cycle-canceling method of Goldberg and Tarjan. The second method is a combination of a cost-scaling technique and a primal network simplex method for the maximum flow problem. We also show that the diameter of the primal network flow polytope is at most n2m.

Operations Research Letters, 1999

In this paper, we present a new polynomial time algorithm for solving the minimum cost network ow problem. This algorithm is based on Edmonds-Karp's capacity scaling and Orlin's excess scaling algorithms. Unlike these algorithms, our algorithm works directly with the given data and original network, and dynamically adjusts the scaling factor between scaling phases, so that it performs at least one ow augmentation in each phase. Our algorithm has a complexity of O(m(m + n log n) log(B=(m + n))), where n is the number of nodes, m is the number of arcs, and B is the sum of the ÿnite arc capacities and supplies in the network.

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.

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.

In this paper we consider the minimum cost network flow problem:

Discrete Applied Mathematics, 1981

If f is a function of several variables, one calls a pair of variables substitutes (co.mp/ements) if the change of the value of the function when both variables are increased is at most (at least) equal to the sum of the changes when each is increased separately. We here consider the case wherefis the value of a maximum weight circulation on a network and the variables are the upper and lower bounds and the wei8hts of a pair of arcs. We introduce a simple combinatorial criterion for two arcs to be in "series" or "parallel" and show that these two cases correspond to the variables being complements or substitutes respectively. This generalizes results of Shapley for the special case of the maximum flow and optimal assignment problems. We also show that our result is best possible in that if two arcs are neither in series nor parallel, then the corresponding variables can be either substitutes or complements or both.

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.