Approximating reduced costs under degeneracy in a network flow problem with side constraints

The problem of optimally (re)allocating Navy personnel to combat units is compounded by several considerations: availability of trained personnel, staffing of positions by occupation groups or ranks, and maintaining an acceptable level of readiness. In this paper we model this problem as a nonlinear nondifferentiable optimization problem. A reformulation of the nonlinear optimization problem as a network flow problem is then developed. The formulation results in a network flow problem with side constraints. An additional, nonnetwork, variable measures the readiness level. This new formulation permits the use of network optimization tools in order to solve effectively very large problems. We then develop two numerical methods for solving this problem. One method is based on a heuristic that solves (approximately) the nondifferentiable problem. The second method is based on a Linear-Quadratic Penalty (LQP) algorithm, and it exploits the embedded network structure by placing the side constraints into the objective function. The resulting nonlinear network program is solved using a simplicial decomposition of the network constraint set. Numerical results indicate the viability of this approach on problems with up to 36,000 arcs and 17,000 nodes with 3,700 side constraints.

This paper we consider graphs with reachability restrictions of three types: barrier, valve and magnetic. As a part of the maximum flow problem in networks with considered restrictions, simple examples are given. These examples demonstrate a significant difference between flows in networks with reachability restrictions and flows in the same networks without any reachability restrictions. In each case under consideration there is a decrease of the maximum flow value. Wherein for magnetic reachability it is shown the possibility to control the network flow, whereas for barrier and valve reachability it is shown the reducing of the maximum flow due to the effect of superimposing of flow to itself, which is never observed in the classical theory of flows in networks.

Report of the Session on Network Flow, Assignment and Travelling Salesman Problems

Network-based formulations of the quadratic assignment problem

It is known that minimum cost flow problems can be solved by successive augmentations along shortest paths. In this paper the issues of implementing shortest path algorithms in this context are examined. Of particular interest is the dynamic topology that the flow networks exhibit. We develop a network generator capable of emulating such topology. Strategies for exploiting the special structures in such networks are discussed. A set of 9000 test problems is offered, from which a particular strategy/algorithm combination is shown to consistently produce superior results when compared to the other combinations.

Purpose. The purpose of the article is to develop a universal unified parallel synchronous algorithm for the implementation of tasks for calculation of maximum one- and multicommodity flows, as well as the creation of a software complex that provides the formation of surface graph models of flows and performs optimal planning of non-uniform flows in transport and other networks. Methodology. The paper investigates the possibilities of previously created and comprehensively verified heuristic parallel synchronous algorithm for calculating maximum one- and multicommodity flows in the networks, establishes its potential limitations, and determines additional advanced procedures that transform the algorithm into a universal parallel algorithm. The proposed parallel synchronous algorithm uses a width-first search strategy while simultaneously identifying possible paths of flows through the network with an estimation of their throughput. Herewith the possibility of analyzing several incremental flows across the network in one iteration was studied. Findings. The article proposes a universal unified parallel synchronous algorithm for calculating maximum flows in networks and develops a unified procedure and software package for planning of non-uniform as well as competitive flows in transport and other networks. The developed software complex implements the problems of formation of surface graph models of networks, for which the problem of optimal planning of non-uniform and competitive multicriteria flows in transport networks is solved. Originality. The article develops a new universal unified parallel synchronous algorithm and procedure for the calculation of optimal uniform, multicommodity and competitive flows in transport networks. Practical value. The practical value of the obtained results is determined by the universal capabilities and efficiency of the procedure for planning non-uniform flows in the networks based on the application of a new parallel synchronous algorithm, as well as the developed software complex, which provides the ability to solve the problems of analysis and planning of uniform and multicommodity flows in transport networks, as well as the implementation of calculation tasks of competitive models of transport and information flows formation. The software complex has a built-in editor of interactive network modeling and a toolbar, which provides both creation of new and downloading existing graphs of networks from the modeling libraries, preservation of optimum flows in the network in the form of an image and a text file, output of errors when working with the program.

The objectives of this chapter are to: Present ideas on graphs and their properties Study flow and potential in networks Discuss the solution procedure of a minimum path problem when all arc lengths are either non-negative or unrestricted in sign Present ideas on network flow, maximum flow problems and generalized maximum flow problems Discuss the source, the sink of a flow, the cut and its capacity in a network flow problem and also the proof of the max-flow min-cut theorem Discuss the solution procedure of maximum flow problems Derive the LPP formulation of a network.

We propose an approach for large-scale non-separable nonlinear multicommodity flow problems by solving a sequence of subproblems which can be addressed by commercial solvers. Using a combination of solution methods such as modified gradient projection, shortest path algorithm and golden section search, the approach can handle general problem instances, including those with (i) non-separable cost, (ii) objective function not available analytically as polynomial but are evaluated using black-boxes, and (iii) additional side constraints not of network flow types. Implemented as a toolbox in commercial solvers, it allows researchers and practitioners, currently conversant with linear instances, to easily manage large-scale convex instances as well. In this article, we compared the proposed algorithm with alternative approaches in the literature, covering both theory and large test cases. New test cases with non-separable convex costs and non-network flow side constraints are also presented and evaluated. The toolbox is available free for academic use upon request.

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are \(\mathcal{NP}\) -hard.

Network flow problems cover a wide range of engineering and management applications. Many streamlined solution methods have been devised for solving different types of the problems. This paper investigates the network flow problems in that the arc lengths of the network are fuzzy numbers. Based on the integer-solution property of the network flow problem, the Yager ranking indices can be calculated for the fuzzy arcs to change the fuzzy formulation of the problem to a crisp formulation. Consequently, the conventional streamlined solution methods can still be applied to find an optimal solution. This optimal solution is proved to be the same as that derived from an exhaustive comparison of all possible solutions. Two examples, one shortest path and one transshipment, discussed in some previous studies illustrate that the method proposed in this paper is able to find the optimal solution. To show that the proposed method is useful in solving real-world problems, the problem of multimedia transmission over the Internet is exemplified.

Tomography is a powerful technique to obtain accurate images of the interior of an object in a nondestructive way. Conventional reconstruction algorithms, such as filtered backprojection, require many projections to obtain high quality reconstructions. If the object of interest is known in advance to consist of only a few different materials, corresponding to known image intensities, the use of this prior knowledge in the reconstruction procedure can dramatically reduce the number of required projections. In previous work we proposed a network flow algorithm for reconstructing a binary image defined on a lattice from its projections. In this paper we propose a new algorithm for the reconstruction of binary images that do not have an intrinsic lattice structure and are defined on a continuous domain, from a small number of their projections. Our algorithm relies on the fact that the problem of reconstructing an image from only two projections can be formulated as a network flow problem in a graph. We derive this formulation for parallel beam and fan beam tomography and show how it can be used to compute binary reconstructions from more than two projections. Our algorithm is capable of computing high quality reconstructions from very few projections. We evaluate its performance on both simulated and real experimental projection data and compare it to other reconstruction algorithms.

We study a type of network flow problem that we call the minimum-cost F-graph flow problem. This problem generalizes the typical minimum-cost network flow problem by allowing the underlying network...

Solving network flow problems is a fundamental component of traffic engineering and many communications applications, such as content delivery or multi-processor scheduling. While a rich body of work has addressed network flow problems in deterministic finding flows in where performance metrics like bandwidth and delay are uncertain and solely known by a probability distribution based on historical data, has received less attention. The work on stochastic networks has predominantly been directed to developing single-path routing algorithms, instead of addressing multi-path routing or flow problems. In this paper, we study constrained maximum flow problems in stochastic networks, where the delay and bandwidth of links are assumed to follow a log-concave probability distribution, which is the case for many distributions that could represent bandwidth and delay. We formulate the maximum-flow problem in such stochastic networks as a convex optimization problem, with a polynomial (in the input) number of variables. When an additional delay constraint is imposed, we show that the problem becomes NP-hard and we propose an approximation algorithm based on convex optimization. Furthermore, we develop a fast heuristic algorithm that, with a tuning parameter, is able to balance accuracy and speed. In a simulation-based evaluation of our algorithms in terms of success ratio, flow values, and running time, our heuristic is shown to give good results in a short running time.

Maximum flow problem on dynamic generative network flows with time-varying bounds

The generation, utilization and disposal of wastes (which constitute environmental hazards) are highlighted in this paper as a network flow problem. In this configuration, we construct and solve as optimal network flow, the problem of minimizing the cost of disposing wastes from sources of generation to dumpsites. Numerical applications based on iterative procedures suggest an appreciable (optimal) reduction in the cost of disposing wastes generated at production sites to various dumpsites. (Journal of Applied Sciences & Environmental Management: 2002 6(1): 64-68)