Equal Flow Problem Research Articles

Algorithms and complexity for the almost equal maximum flow problem

AbstractIn the equal maximum flow problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set, so called homologous arc sets. In this article, we study the closely related almost equal maximum flow problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called deviation function . We prove that the integer AEMFP is in general ‐complete, and show that even the problem of finding a fractional maximum flow in the case of convex deviation functions is also ‐complete. This is in contrast to the EMFP, which is polynomial time solvable in the fractional case. Additionally, we provide inapproximability results for the integral AEMFP. For the (fractional) concave AEMFP we state a strongly polynomial algorithm for the linear and concave piecewise polynomial deviation function case for a fixed number of homologous sets using a parametric search approach.

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.

A Network Simplex Algorithm for the Equal Flow Problem on a Generalized Network

A network simplex algorithm is described for the minimum-cost network flow problem on a generalized network, with the additional constraint that there exist sets of arcs that must carry equal amounts of flow. This problem can be modeled as a linear programming problem and solved using the standard simplex algorithm. However, because of the structure of the problem, more efficient algorithms are possible that solve the problem by operating directly on the network itself. One such algorithm is described that leads to improved asymptotic performance per iteration over the standard simplex algorithm, as long as the number of side constraints is small relative to the size of the network. Computational results are given comparing this algorithm to CPLEX's primal simplex solver on randomly generated graphs.

Integer equal flows

The integer equal flow problem is an NP-hard network flow problem, in which all arcs in given sets R 1 , … , R ℓ must carry equal flow. We show that this problem is effectively inapproximable, even if the cardinality of each set R k is two. When ℓ is fixed, it is solvable in polynomial time.

Network simplex algorithm for the general equal flow problem

In this paper the general equal flow problem is considered. This is a minimum cost network flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the same value. This model can be applied to approach water resource system management problems or multiperiod logistic problems in general involving policy restrictions which require some arcs to carry the same amount of flow through the given study period. Although the bases of the general equal flow problem are no longer spanning trees, it is possible to recognize a similar structure that allows us to take advantage of the practical computational capabilities of network models. After characterizing the bases of the problem as good ( r+1)-forests, a simplex primal algorithm is developed that exploits the network structure of the problem and requires only slight modifications of the well-known network simplex algorithm.

Algorithms for the Simple Equal Flow Problem

The minimum cost flow problem is to determine a least cost shipment of a commodity through a network G = (N, A) in order to satisfy demands at certain nodes from available supplies at other nodes. In this paper, we study a variant of the minimum cost flow problem where we are given a set R ⊆ A of arcs and require that each arc in R must carry the same amount of flow. This problem, which we call the simple equal flow problem, arose while modeling a water resource system management in Sardinia, Italy. We consider the simple equal flow problem in a directed network with n nodes, m arcs, and where all arc capacities and node supplies are integer and bounded by U. We develop several algorithms for the simple equal flow problem—the network simplex algorithm, the parametric simplex algorithm, the combinatorial parametric algorithm, the binary search algorithm, and the capacity scaling algorithm. The binary search algorithm solves the simple equal flow problem in O(log(nU)) applications of any minimum cost flow algorithm. The capacity scaling algorithm solves it in O(m(m + n logn) log (nU)) time, which is almost the same time needed to solve the minimum cost flow problem by the capacity scaling algorithm. These algorithms can be easily modified to obtain an integer solution of the simple equal flow problem.

An efficient lagrangean relaxation scheme for linear and integer equal flow problems

The Equal flow problem is a minimal cost network flow problem with side constraints which state that the amounts of flows on certain pairs of arcs shall coincide. We present a heuristic algorithm for linear as well as integer equal flow problems; it uses Lagrangean dualization with respect to the equal flow side constraints and subgradient optimization to solve the Lagrangean dial problem which provides lower bounds on the optimal value of the original problem. Embedded in the subgradient procedure is a simple computation which, in the limit, will produce optimal values of the equal flow variables of the linear problem. A heuristic procedure for finding feasible solutions to the equal flow problem, and upper bounds on its optimal value, is obtained by fixing the equal flow variables to the near-optimal values that are a available in each iteration and solving the resulting minimal cost network flow problems, which involve the ordinary variables only. The algorithm has been implemented for solving equal flow transportation and transshipment problems, and the computational results indicate that it is highly efficient

A Primal Simplex Specialization for the Equal Flow Problem

Abstract This paper presents a primal simplex specialization for the equal flow problem. This approach is motivated by a desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The simplex computations are discussed and a starting solution based on the subgradient optimization technique is presented. Our implementation is tested on problems with up to 1500 nodes, 6600 arcs and 600 equal flow pairs. These problems are also solved using MPSX and another specialized code for the equal flow problem. We present Computational experience which indicates that our approach is well suited for problems with up to 150 side constraints. As the number of pairs increased, MPSX with an advanced start procedure, performed better than our specialized code.

The equal flow problem

This paper presents a new algorithm for the solution of a network problem with equal flow side constraints. The solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The proposed algorithm makes use of Lagrangean relaxation to obtain a lower bound and decomposition by right-hand-side allocation to obtain upper bounds. The lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound, but also allows an initial allocation of equal flows for the upper bound. The algorithm has been tested on problems with up to 1500 nodes and 6000 arcs. Computational experience indicates that solutions whose objective function value is well within 1% of the optimum can be obtained in 1%–65% of the MPSX time depending on the amount of imbalance inherent in the problem. Incumbent integer solutions which are within 99.99% feasible and well within 1% of the proven lower bound are obtained in a straightforward manner requiring, on the average, 30% of the MPSX time required to obtain a linear optimum.