Generalized Network Flow Problems Research Articles

Novel GPU-Based Method for the Generalized Maximum Flow Problem

This paper investigates the application of a minimum loss path finding algorithm to determine the maximum flow in generalized networks that are characterized by arc losses or gains. In these generalized network flow problems, each arc has not only a defined capacity but also a loss or gain factor, which must be taken into consideration when calculating the maximum achievable flow. This extension of the traditional maximum flow problem requires a more comprehensive approach, where the maximum amount of flow is determined by accounting for additional factors such as costs, varying arc capacities, and the specific loss or gain associated with each arc. This paper extends the classic Ford–Fulkerson algorithm, adapting it to iteratively identify source-to-sink (s − t) residual directed paths with minimum cumulative loss and generalized augmenting paths (GAPs), thus enabling the efficient computation of maximum flow in such complex networks. Moreover, to enhance the computational performance of the proposed algorithm, we conducted extensive studies on parallelization techniques using graphics processing units (GPUs). Significant improvements in the algorithm’s efficiency and scalability were achieved. The results demonstrate the potential of GPU-accelerated computations in handling real-world applications where generalized network flows with arc losses and gains are prevalent, such as in telecommunications, transportation, or logistics networks.

Fast and Secure Routing Algorithms for Quantum Key Distribution Networks

We investigate the problem of fast and secure packet routing in multi-hop Quantum Key Distribution (QKD) networks. We consider a practical trusted-node setup where a QKD protocol randomly generates symmetric private key pairs over each QKD-enabled link in a network. Packets are first encrypted with the available quantum keys and then transmitted on a point-to-point basis. A fundamental problem in this setting is the design of a secure and capacity-achieving routing policy that takes into account the time-varying availability of the encryption keys and diverse physical-layer link capacities. To address this problem, we propose a new secure throughput-optimal policy called Tandem Queue Decomposition (TQD). The TQD policy is designed by incorporating the QKD process into the Universal Max Weight (UMW) routing policy. We show that the TQD policy achieves the entire secure capacity region for a broad class of traffic, including unicast, broadcast, multicast, and anycast. The TQD policy operates by reducing the problem to the generalized network flow problem without the key availability constraints over a transformed network. The throughput-optimality of the TQD policy is established using the Lyapunov stability theory by carefully analyzing the interdependent queueing process and the key-storage dynamics. Finally, we demonstrate the practical efficiency of the TQD policy over the existing routing algorithms by numerically comparing their performance on a realistic simulator built on top of the state-of-the-art OMNeT++ network simulator platform.

Optimal Control for Generalized Network-Flow Problems

We consider the problem of throughput-optimal packet dissemination, in the presence of an arbitrary mix of unicast, broadcast, multicast, and anycast traffic, in an arbitrary wireless network. We propose an online dynamic policy, called Universal Max-Weight (UMW), which solves the problem efficiently. To the best of our knowledge, UMW is the first known throughput-optimal policy of such versatility in the context of generalized network flow problems. Conceptually, the UMW policy is derived by relaxing the precedence constraints associated with multi-hop routing and then solving a min-cost routing and max-weight scheduling problem on a virtual network of queues . When specialized to the unicast setting, the UMW policy yields a throughput-optimal cycle-free routing and link scheduling policy. This is in contrast with the well-known throughput-optimal back-pressure (BP) policy which allows for packet cycling, resulting in excessive latency. Extensive simulation results show that the proposed UMW policy incurs a substantially smaller delay as compared with the BP policy. The proof of throughput-optimality of the UMW policy combines ideas from the stochastic Lyapunov theory with a sample path argument from adversarial queueing theory and may be of independent theoretical interest.

Convexification of generalized network flow problem

This paper is concerned with the minimum-cost flow problem over an arbitrary flow network. In this problem, each node is associated with some possibly unknown injection and each line has two unknown flows at its ends that are related to each other via a nonlinear function. Moreover, all injections and flows must satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which is shown to always obtain globally optimal injections. This relaxation may fail to find optimal flows because the mapping from injections to flows is not unique in general. We show that the proposed relaxation, named convexified GNF (CGNF), obtains a globally optimal flow vector if the optimal injection vector is a Pareto point. More generally, the network can be decomposed into two subgraphs such that the lines between the subgraphs are congested at optimality and that CGNF finds correct optimal flows over all lines of one of these subgraphs. We also fully characterize the set of all globally optimal flow vectors, based on the optimal injection vector found via CGNF. In particular, we show that this solution set is a subset of the boundary of a convex set, and may include an exponential number of disconnected components. A primary application of this work is in optimization over electrical power networks.

Trust Evaluation in Online Social Networks Using Generalized Network Flow

In online social networks (OSNs), to evaluate trust from one user to another indirectly connected user, the trust evidence in the trusted paths (i.e., paths built through intermediate trustful users) should be carefully treated. Some paths may overlap with each other, leading to a unique challenge of path dependence , i.e., how to aggregate the trust values of multiple dependent trusted paths. OSNs bear the characteristic of high clustering, which makes the path dependence phenomenon common. Another challenge is trust decay through propagation, i.e., how to propagate trust along a trusted path, considering the possible decay in each node. We analyze the similarity between trust propagation and network flow, and convert a trust evaluation task with path dependence and trust decay into a generalized network flow problem. We propose a modified flow-based trust evaluation scheme GFTrust , in which we address path dependence using network flow, and model trust decay with the leakage associated with each node. Experimental results, with the real social network data sets of Epinions and Advogato, demonstrate that GFTrust can predict trust in OSNs with a high accuracy, and verify its preferable properties.

Energy-Saving Generation Dispatch Using Minimum Cost Flow

This study uses a minimum cost flow method to solve a dispatch problem in order to minimize the consumption of coal in the dispatching of a thermal power system. Low-carbon generation dispatching is also considered here since the scheduling results are consistent with energy-saving generation dispatch. Additionally, this study employs minimum coal consumption as an objective function in considering the output constraints, load balance constraints, line loss, ramp rate limits, spinning reverse needs, prohibited operating zone requirements, security constraints, and other common constraints. The minimum cost flow problem, considering the loss of network flow, is known as a generalized network flow problem, which can be expressed as a quadratic programming problem in mathematics. Accordingly, the objective function was solved by LINGO11, which was used to calculate a network in a single time; a continuous period dispatch plan was obtained by accumulating each period network flow together. This analysis proves it feasible to solve a minimal cost flow problem with LINGO11. Theoretical analysis and numerical results prove the correctness and effectiveness of the proposed method.

Multichannel cooperative sensing for cognitive radio with users owning heterogeneous sensing ability

SUMMARYMultichannel cooperative sensing (MCS) is an effective method for dynamic spectrum access in cognitive radio networks. In contrast to most existing work on MCS that considered secondary users with homogeneous sensing ability, this paper studies the MCS problem for secondary users with heterogeneous sensing ability in terms of sensing accuracy. We further take into account different parameters of primary channels such as bandwidth, probability of being idle in each sensing period, and frequency selective fading at the sensing receiver. The MCS problem is formulated as a weapon target assignment problem, where more sensing resources are assigned to channels that are more valuable. This weapon target assignment problem is transformed to an integer generalized network flow problem with convex flow costs to obtain the lower bound solution, and then solved by the branch and bound algorithm with this bound to yield the exact scheme. To reduce computational complexity, a heuristic scheme is also proposed, which has approximate performance compared with the exact scheme. Finally, extensive simulation results for different scenarios illustrate the performance improvements of the proposed schemes over the existing scheme. Copyright © 2013 John Wiley & Sons, Ltd.

A branch‐and‐bound algorithm for representative integer efficient solutions in multiple objective network programming problems

AbstractIn many applications of multiple objective network programming (MONP) problems, only integer solutions are acceptable as the final optimal solution. Representative efficient solutions are usually obtained by sampling the efficient set through the solution of augmented weighted Tchebycheff network programs. Because such efficient solutions are usually not integer solutions, a branch‐and‐bound (BB) algorithm is developed to find integer efficient solutions. The purpose of the BB algorithm is to support interactive procedures by generating representative integer efficient solutions. To be computationally efficient, the algorithm takes advantage of the network structure as much as possible. An algorithm, used in the BB algorithm and performed on the key tree, is developed to construct feasible solutions from infeasible solutions and basic solutions from nonbasic solutions when bounds on branching variables change. The BB algorithm finds basic and nonbasic or supported and unsupported integer efficient solutions as long as they are optimal. Details of the algorithm are presented, an example is provided and computational results are reported. Computational results show that the BB algorithm performs well. Although the BB algorithm is developed for the purpose of generating integer efficient solutions for MONP problems, it can also solve more general integer network flow problems with linear side constraints. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Space Optimization for Warehousing Problem: A Methodology for Decision Support System

This article presents a way of tackling a special class of space optimization problems that arise in a number of practical applications in industry and elsewhere. It presents an elegant solution to a problem that was considered by (Das, 2005) in optimizing storage space in warehouse of a footwear manufacturing company. In (Das, 2005), the problem was formulated as a nonlinear programming problem. In this article, it is shown that the problem can be formulated as a generalized transportation problem which is a special case of generalized network flow problems. Further, an elegant scheme is devised to handle the dynamic situation of warehousing problem which can be easily translated into a decision support system for the warehouse management system. Also, the article points out certain obscurities and gaps in (Das, 2005).

Heuristic solutions for general concave minimum cost network flow problems

AbstractWe address the single‐source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of problems in their full generality are only able to address small to medium size instances, since this class of problems is known to be NP‐Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test problems are compared to optimal solutions obtained by a dynamic programming method for the smaller problem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 67–76 2007

Optimal reconfiguration of satellite constellations with the auction algorithm

Abstract Traditionally, satellite constellation design has focused on optimizing global, zonal or regional coverage with a minimum number of satellites. In some instances, however, it is desirable to deploy a constellation in stages to gradually expand capacity. This requires launching additional satellites and reconfiguring the existing on-orbit satellites. Also, a constellation might be retasked and reconfigured after it is initially fielded for operational reasons. This paper presents a methodology for optimizing orbital reconfigurations of satellite constellations. The work focuses on technical aspects for transforming an initial constellation A into a new constellation, B , typically with a larger number of satellites. A general framework was developed to study the orbital reconfiguration problem. The framework was applied to low Earth orbit constellations of communication satellites. This paper specifically addresses the problem of determining the optimal assignment for transferring on-orbit satellites in constellation A to constellation B such that the total Δ V for the reconfiguration is minimized. It is shown that the auction algorithm, used for solving general network flow problems, can efficiently and reliably determine the optimum assignment of satellites of A to slots of B . Based on this methodology, reconfiguration maps can be created, which show the energy required for transforming one constellation into another as a function of type (Street-of-Coverage, Walker, Draim), altitude, ground elevation angle and fold of coverage. Suggested extensions of this work include quantification of the tradeoff between reconfiguration time and Δ V , multiple successive reconfigurations, balancing propellant consumption within the constellation during reconfiguration as well as using reconfigurability as an objective during initial constellation design.

Generalized Networks: The Theory of Preprocessing and an Empirical Analysis

The idea of preprocessing a linear program with the goal of eliminating constraints or columns is well established. The default setting for CPLEX is to apply preprocessing prior to application of any of its four solvers. In this investigation, a set of preprocessing procedures is developed for the special case of a generalized network flow problem. Some procedures involve adding or replacing arcs to eliminate a node. These procedures all produce an optimal basic feasible solution and optimal dual solution to the original problem. In an empirical analysis, our generalized network preprocessor ran approximately seven times faster than the CPLEX 6.5.3 preprocessor, resulting in an overall solution time reduction of over 20%.

Monotonizing linear programs with up to two nonzeroes per column

Linear programming problems with up to two nonzeroes per column in the constraint matrix are shown equivalent to generalized network flow problem. The transformation is applied for solving the maximum cut problem, the b-matching problem in strongly polynomial time and for approximation algorithms for certain integer versions of the problem.

Process selection and sequencing in a two-agents production system

This paper addresses a coordination problem concerning two production agents in a manufacturing system. The two agents have a set of processes which must be carried out on some common resource. They can use the resource individually, but in general there may be an overall advantage in concurrently performing certain processes. The problem is to select concurrent processes and to sequence them with the aim of minimizing the total number of set-ups. In particular, a decomposition approach is proposed, in which first the concurrent processes are selected, by solving a generalized network flow problem, and then an optimal process sequencing is efficiently found.

Combinatorial interior point methods for generalized network flow problems

We present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on Wallacher and Zimmermann's combinatorial interior point method for the minimum cost network flow problem. The algorithms have features of both a combinatorial algorithm and an interior point method. They work towards optimality by iteratively reducing the value of a potential function while maintaining interior point solutions. At each iteration, flow is augmented along a generalized circulation, which is computed by solving a TVPI (Two Variables Per Inequality) system. The algorithms run in time, where m and n are, respectively, the number of arcs and nodes in the graph, and L is the length of the input data.

Implementation and Test of Auction Methods for Solving Generalized Network Flow Problems with Separable Convex Cost

We describe the implementation and testing of two methods, based on the auction approach, for solving the problem of minimizing a separable convex cost subject to generalized network flow conservation constraints. The first method is the e-relaxation method of Ref. 1; the second is an extension of the auction sequential/shortest path algorithm for ordinary network flow to generalized network flow. We report test results on a large set of randomly generated problems with varying topology, arc gains, and cost function. Comparison with the commercial code CPLEX on linear/quadratic cost problems and with the public-domain code PPRN on nonlinear cost ordinary network problems are also made. The test results show that the auction sequential/shortest path algorithm is generally fastest for solving quadratic cost problems and mixed linear/nonlinear cost problems with arc gain range near 1. The e-relaxation method is generally fastest for solving nonlinear cost ordinary network problems and mixed linear/nonlinear cost problems with arc gain range away from 1. CPLEX is generally fastest for solving linear cost and mixed linear/quadratic cost problems with arc gain range near 1.

Combinatorial Approximation Algorithms for Generalized Flow Problems

Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier ?((v,w)) for each arc (v,w). For every unit of flow entering the arc, ?((v,w)) units of flow exit. We present a strongly polynomial algorithm for a single-source generalized shortest paths problem, using a left-distributive closed semiring. This permits use of a dynamic-programming routine similar to the Bellman?Ford algorithm, given a guess for the value of the optimal solution. Using Megiddo's parametric search scheme, we can compute the optimal value in strongly polynomial time. The algorithm's running time O(mn2logn) matches the previously best known, but the algorithm is simpler, is based on the well-known theory of closed semirings, and directly works with the given graph. All previous polynomial-time algorithms were based on interior-point methods or directly solved the dual problem and translated the solution back to the primal problem. Using this generalized shortest paths algorithm, we present fully polynomial-time approximation schemes for the generalized versions of the maximum flow, the nonnegative-cost minimum-cost flow, the concurrent flow, the multicommodity maximum flow, and the multicommodity nonnegative-cost minimum-cost flow problems with running times independent of the size of the flow multipliers' representation.

An ε-relaxation method for separable convex cost generalized network flow problems

Cover title. The extended abstract of this article appeared in the proceedings of the 5th International IPCO Conference, Vancouver, June 1996.--P. 1.

Faster Algorithms for the Generalized Network Flow Problem

We consider the generalized network flow problem. Each arc e in the network has a gain factor γ(e). If f(e) units of flow enter arc e, then f(e)γ(e) units arrive at the other end of e. The objective is to maximize the net flow into one specific node, the sink. The constraints are the conservation of flow at nodes and the capacities of arcs. We present a combinatorial algorithm which solves this problem in Õ(m2(m + n log log B)log B) time, where n is the number of nodes, m is the number of arcs, and B is the largest integer used to represent the gain factors, the capacities, and the initial supplies at the nodes. If m is O(n4/3−ε) and B is not extremely large, then our bound is better than the previous best upper bound for this problem. We also improve the best known upper bound for the approximate generalized flow problem by showing that a solution whose value is within a factor of 1 + ξ from the optimum can be computed in Õ(m(m + n log log B)log (1/ξ)) time plus Õ(mn2 log B) time for preprocessing.

An intelligent modeling system for generalized network flow problems: With application to planning for multinational firms

This paper presents a discussion of the Generalized Network System (GNS), a system that captures knowledge about generalized network flow problems, in order to help users formulate, solve and interpret generalized network flow problems. Although previous researchers have built intelligent systems that incorporate knowledge about linear programming, this system includes more specific knowledge about generalized network flow problems. GNS is illustrated using it in a setting that requires international financial and replan rapidly. In addition, they need to be able to model complex events and organization GNS is illustrated using it in a setting that requires international financial and replan rapidly. In addition, they need to be able to model complex events and organization structures. For example, multinational planners need to be able to plan for production in multiple countries and repatriatization of funds. GNS allows users to meet these needs.