Robust minimum cost flow problem under consistent flow constraints

The robust minimum cost flow problem under consistent flow constraints (RobMCF$$\equiv $$ ≡ ) is a new extension of the minimum cost flow (MCF) problem. In the RobMCF$$\equiv $$ ≡ problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a finite discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem in networks with integral demands, supplies, and capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCF$$\equiv $$ ≡ problem is strongly $$\mathcal {NP}$$ NP -hard on acyclic digraphs by a reduction from the (3, B2)-Sat problem. Further, we demonstrate that the RobMCF$$\equiv $$ ≡ problem is weakly $$\mathcal {NP}$$ NP -hard on series-parallel digraphs by providing a reduction from Partition. If in addition the number of scenarios is constant, we propose a pseudo-polynomial algorithm based on dynamic programming. Finally, we present a special case on series-parallel digraphs for which we can solve the RobMCF$$\equiv $$ ≡ problem in polynomial time.

Efficient Algorithms for Identifying Loop Formation and Computing θ Value for Solving Minimum Cost Flow Network Problems

While the minimum cost flow (MCF) problems have been well documented in many publications, due to its broad applications, little or no effort have been devoted to explaining the algorithms for identifying loop formation and computing the θ value needed to solve MCF network problems. This paper proposes efficient algorithms, and MATLAB computer implementation, for solving MCF problems. Several academic and real-life network problems have been solved to validate the proposed algorithms; the numerical results obtained by the developed MCF code have been compared and matched with the built-in MATLAB function Linprog() (Simplex algorithm) for further validation.

An Algorithm for Choosing, Ordering a New Criteria of a Bi-Objective Flow Problem

In this paper, we propose an algorithm which is based on many things: the notions well-known of the simplex network method, Ford Fulkerson’s algorithm and our new idea, which is << the gain cycles >>, applied on a bi-objective minimum cost flow problem. This algorithm permits us to have a good order of many criteria in a rapid and an efficient way; because this classification permits us to structure the optimal area, in which we can choose the best action among the others which exist in the objective space. From this one, we distinguish, that the resolution of this problem comes to find an under set of good actions, among which the decider can select an action of best compromise, or make a decision, in the case where reference indications of the deciders may change. A didactic example is done to illustrate our algorithm.

Solving Robust Variants of Integer Flow Problems with Uncertain Arc Capacities

This paper deals with robust optimization and network flows. Several robust variants of integer flow problems are considered. They assume uncertainty of network arc capacities as well as of arc unit costs (where applicable). Uncertainty is expressed by discrete scenarios. Since the considered variants of the maximum flow problem are easy to solve, the paper is mostly concerned with NP-hard variants of the minimum-cost flow problem, thus proposing an approximate algorithm for their solution. The accuracy of the proposed algorithm is verified by experiments.

Investigation and modification of the inversed vortex phase field method for phase unwrapping

The paper focuses on the problem of the phase unwrapping in spaceborne remote-sensing interferometric synthesized aperture radar (InSAR) systems. Major unwrapping methods and techniques are considered and the modification of the inversed vortex phase field method of phase unwrapping for interferometric data processing of space-borne synthesized aperture radars is proposed. The modification includes the separation and unwrapping of the low-frequency phase only, and obtaining of the residual phase interferogram, which phase range does not exceed 1-2 ambiguity height values. This approach significantly reduces the number of phase residues and increases the processing speed. The other modification implies filter processing of the residual phase without phase unwrapping, which includes iterative separation of the low-frequency using the Gaussian filter and phase subtraction. This approach moves phase fringes to the relief inflection areas, and is similar to the minimum-cost flow unwrapping results. The computational complexity of the algorithm is proportional to the interferogram size and the number of the phase residues of the low-frequency phase interferogram. The accuracy of digital elevation models obtained by the algorithm was estimated using the ALOS PALSAR radar data and the reference altitude data. The results show, that the accuracy is compared with the minimum-cost flow method, but has the less computational complexity.