A Primal Algorithm for Finding Minimum-Cost Flows in Capacitated Networks With Applications (original) (raw)

Polynomial-time primal simplex algorithms for the minimum cost network flow 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.

Minimum cost network flows: Problems, algorithms, and software

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.

An Application of Network Simplex Method for Minimum Cost Flow Problems

Networks are more convenient for modeling because of their simple mathematical structure that can be easily represented with a graph. This simplicity takes an advantage with regard to algorithmic efficiency. In this paper, an implementation of network simplex algorithm is described for solving the minimum cost network flow problem which is one of the most fundamental and significant problems in the optimal design on a generalized network with the additional constraint. Network flow problem can be defined by a given set of nodes and arcs with known cost parameters for each arc and fixed external flow for each node. The optimization problem is to send flow from a set of supply nodes, through the arcs of a network, to a set of demand nodes, at minimum total cost subject to the arc capacity constraints. The simplex algorithm applied to the network flow programming problem. Network simplex method describes basic solutions for the network flow programming problem and provides procedures for computing the primal and dual solutions associated with a given basis to find the optimal solution.

A Lagrangean heuristic for the capacitated concave minimum cost network flow problem

European Journal of Operational Research, 1994

We propose a heuristic solution technique for the capacitated concave minimum cost network flow problem based on a Lagrangean dualization of the problem. Despite its dual character the algorithm guarantees the generation of primal feasible solutions which are local optima and therefore candidates of being the global optimum. The Lagrangean dual is solved by a subgradient search procedure and provides a lower bound to the optimal value. The lower bound is, in general, stronger than the one obtained by a linear approximation of the original problem. It can be used as a judgement of the quality of the solution or in a branch and bound procedure. Computational results from randomly generated problems are presented.

A polynomial time primal network simplex algorithm for minimum cost flows

Mathematical Programming, 1997

Developing a polynomial time algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2 m log nC, n 2 m 2 log n)) time, where n is the number of nodes in the network, m is the number of arcs, and C denotes the maximum absolute arc costs if arc costs are integer and 0 otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the "premultiplier algorithm." A vector X of node potentials is called a vector of premultipliers with respect to a rooted tree if each arc directed towards the root has a non-positive reduced cost and each arc directed away from the root has a non-negative reduced cost. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm log nC, nm 2 log n)) pivots, With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm log n).

Algorithms for the single-source uncapacitated minimum concave-cost network flow problem

Journal of Global Optimization, 1991

We investigate algorithms, applications, and complexity issues for the single-source uncapacitated (SSU) version of the minimum concave-cost network flow problem (MCNFP). We present applications arising from production planning, and prove complexity results for both global and local search. We formally state the local search algorithm of Gallo and Sodini [5], and present alternative local search algorithms. Computational results are provided to compare the various local search algorithms proposed and the effects of initial solution techniques.

An exterior Simplex type algorithm for the minimum cost network flow problem

Computers & Operations Research, 2009

In this paper a new Network Exterior Point Simplex Algorithm (NEPSA) for the Minimum Cost Network Flow Problem (MCNFP) is analytically presented. NEPSA belongs to a special simplex type category and is a modification of the classical network simplex algorithm. The main idea of the algorithm is to compute two flows. One flow is basic but not always feasible and the other is feasible but not always basic. A complete proof of correctness for the proposed algorithm is also presented. Moreover, the computational behavior of NEPSA is shown by an empirical study carried out for randomly generated sparse instances created by the well known gridgen network problem generator.

A specialized network simplex algorithm for the constrained maximum flow problem

European Journal of Operational Research, 2011

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.