A Note on Feasible Flows in Lower-Bounded Multicommodity Networks

This note considers the feasibility for two types of multicommodity flow problems: maximal flow problems with both upper and lower arc capacities, and capacitated minimal cost trans-shipment problems. Although closed form conditions analogous to those known for single commodity problems cannot be derived, it is shown that feasibility is equivalent to finding a maximal multicommodity flow of a specified value on a related network with zero lower bounds, a direct extension of well-known results for single commodity networks.     

Constraint Generation for Network Reliability Problems

This paper presents a constraint generation approach to the network reliability problem of adding spare capacity at minimum cost that allows the traffic on a failed link to be rerouted to its destination. Any number of non-simultaneous link failures can be part of the requirements on the spare capacity. The key result is a necessary and sufficient condition for a multicommodity flow to exist, which is derived in the appendix. Computational results on large numbers of random networks are presented.     

Approximate decomposition methods for the analysis of multicommodity flow routing in generalized queuing networks

In this study we deal with network routing decisions and approximate performance evaluation approaches for generalized open queuing networks (OQN), in which commodities enter the network, receive service at one or more arcs and then leave the network. Exact performance evaluation has been applied for the analysis of Jackson OQN, where the arrival and service processes of the commodities are assumed to be Poisson. However, the Poisson processes’ hypotheses are not a plausible or acceptable assumption for the analysis of generalized OQN, as their arrival and service processes can be much less variable than Poisson processes, resulting in overestimated system performance measures and inappropriate flow routing solutions. In this paper we merge network routing algorithms and network decomposition methods to solve multicommodity flow problems in generalized OQN. Our focus is on steady-state performance measures as average delays and waiting times in queue. The main contributions are twofold: (i) to highlight that solving the corresponding multicommodity flow problem by representing the generalized OQN as a Jackson OQN may be a poor approximation and may lead to inaccurate estimates of the system performance measures, and (ii) to present a multicommodity flow algorithm based on a routing step and on an approximate decomposition step, which leads to much more accurate solutions. Computational results are presented in order to show the effectiveness of the proposed approach.     

Discrete Cost Multicommodity Network Optimization Problems and Exact Solution Methods

We first introduce a generic model for discrete cost multicommodity network optimization, together with several variants relevant to telecommunication networks such as: the case where discrete node cost functions (accounting for switching equipment) have to be included in the objective; the case where survivability constraints with respect to single-link and/or single-node failure have to be taken into account. An overview of existing exact solution methods is presented, both for special cases (such as the so-called single-facility and two-facility network loading problems) and for the general case where arbitrary step-increasing link cost-functions are considered. The basic discrete cost multicommodity flow problem (DCMCF) as well as its variant with survivability constraints (DCSMCF) are addressed. Several possible directions for improvement or future investigations are mentioned in the concluding section.     

Algorithms for multiplayer multicommodity flow problems

We investigate the multiplayer multicommodity flow problem: several players have different networks and commodities over a common node set. Pairs of players have contracts where one of them agrees to route the flow of the other player (up to a given capacity) between two specified nodes. In return, the second player pays an amount proportional to the flow value. We show that the social optimum can be computed by linear programming, and we propose algorithms based on column generation and Lagrangian relaxation. In contrast, we prove that it is hard to decide if an equilibrium solution exists, although some natural conditions guarantee its existence.     

New formulation and relaxation to solve a concave-cost network flow problem

In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.     

具有全局性公平满意度的最大多物资网络流问题

我们研究-个具有全局性公平满意度的最大多物资网络流问题(MMFP-GFMR).该项工作不仅丰富了最大多物资网络流问题的内容,而且可用于研究某些实际优化决策问题,例如运输过程中的一些资源分配问题.文中主要内容如下:(A)定义问题MMFP-GFMR并证明其解的存在性.(B)设计-个求解MMFP-GFMR的拟多项式逼近算法.(C)研究算法的复杂性与逼近程度.(D)最后通过模拟计算验证了我们的工作.     

Fast and simple approximation schemes for generalized flow

We present fast and simple fully polynomial-time approximation schemes (FPTAS) for generalized versions of maximum flow, multicommodity flow, minimum cost maximum flow, and minimum cost multicommodity flow. We extend and refine fractional packing frameworks introduced in FPTAS’s for traditional multicommodity flow and packing linear programs. Our FPTAS’s dominate the previous best known complexity bounds for all of these problems, some by more than a factor of n ², where n is the number of nodes. This is accomplished in part by introducing an efficient method of solving a sequence of generalized shortest path problems. Our generalized multicommodity FPTAS’s are now as fast as the best non-generalized ones. We believe our improvements make it practical to solve generalized multicommodity flow problems via combinatorial methods. Received: June 3, 1999 / Accepted: May 22, 2001?Published online September 17, 2001     

Designing WDM Optical Networks Using Branch-and-Price

In this paper, we present an exact solution procedure for the design of two-layer wavelength division multiplexing (WDM) optical networks with wavelength changers and bifurcated flows. This design problem closely resembles the traditional multicommodity flow problem, except that in the case of WDM optical networks, we are concerned with the routing of multiple commodities in two network layers. Consequently, the corresponding optimization models have to deal with two types of multicommodity variables defined for each of the network layers. The proposed procedure represents one of the first branch-and-price algorithms for a general WDM optical network setting with no assumptions on the number of logical links that can be established between nodes in the network. We apply our procedure in a computational study with four different network configurations. Our results show that for the three tested network configurations our branch-and-price algorithm provides solutions that are on average less than 5 % from optimality. We also provide a comparison of our branch-and-price algorithm with two simple variants of the upper bounding heuristic procedure HLDA that is commonly used for WDM optical network design.     

An implementation of linear and nonlinear multicommodity network flows

This work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities. For the multicommodity network flow problem through primal partitioning the code implements a specialization of Murtagh and Saunders' strategy of dividing the set of variables into basic, nonbasic and superbasic. Several tests are reported, using random problems obtained from different network generators and real problems arising from the fields of long and short-term hydrothermal scheduling of electricity generation and traffic assignment, with sizes of up to 150000 variables and 45 000 constraints The performance of the code developed is compared to that of alternative methodologies for solving the same problems: a general purpose linear and nonlinear constrained optimization code, a specialised linear multicommodity network flow code and a primal-dual interior point code.     

有预算限制的最大多种物资流问题

研究有预算限制的最大多种物资流问题,给出了这个问题的不依赖物资数k的全多项式时间近似算法,其算法复杂性是O~(-ε2m2).同时,利用有预算限制的最大多种物资流问题的研究结果,我们也得到了费用最小的最大多种物资流问题的近似算法和算法复杂性.     

Multicommodity flows in cycle graphs

This paper considers the multicommodity flow problem and the integer multicommodity flow problem on cycle graphs. We present two linear time algorithms for solving each of the two problems.     

A cycle augmentation algorithm for minimum cost multicommodity flows on a ring

Minimum cost multicommodity flows are a useful model for bandwidth allocation problems. These problems are arising more frequently as regional service providers wish to carry their traffic over some national core network. We describe a simple and practical combinatorial algorithm to find a minimum cost multicommodity flow in a ring network. Apart from 1 and 2-commodity flow problems, this seems to be the only such “combinatorial augmentation algorithm” for a version of exact mincost multicommodity flow. The solution it produces is always half-integral, and by increasing the capacity of each link by one, we may also find an integral routing of no greater cost. The “pivots” in the algorithm are determined by choosing an >0, increasing and decreasing sets of variables, and adjusting these variables up or down accordingly by . In this sense, it generalizes the cycle cancelling algorithms for (single source) mincost flow. Although the algorithm is easily stated, proof of its correctness and polynomially bounded running time are more complex.     

广义多品种最小费用流问题的对偶理论

基于广义多品种最小费用流问题的性质，将问题转化成一对含有内、外层问题的双水平规划，内层规划实际是单品种费用流问题，而外层问题是分离的凸规划，使用相关的凸分析理论，导出了广义多品种最小费用流问题的对偶规划，对偶定理和Kuhn-Tucker条件。     

Analyzing quadratic unconstrained binary optimization problems via multicommodity flows

Quadratic Unconstrained Binary Optimization (QUBO) problems concern the minimization of quadratic polynomials in n{0,1}-valued variables. These problems are NP-complete, but prior work has identified a sequence of polynomial-time computable lower bounds on the minimum value, denoted by C₂,C₃,C₄,…. It is known that C₂ can be computed by solving a maximum flow problem, whereas the only previously known algorithms for computing require solving a linear program. In this paper we prove that C₃ can be computed by solving a maximum multicommodity flow problem in a graph constructed from the quadratic function. In addition to providing a lower bound on the minimum value of the quadratic function on {0,1}ⁿ, this multicommodity flow problem also provides some information about the coordinates of the point where this minimum is achieved. By looking at the edges that are never saturated in any maximum multicommodity flow, we can identify relational persistencies: pairs of variables that must have the same or different values in any minimizing assignment. We furthermore show that all of these persistencies can be detected by solving single-commodity flow problems in the same network.     

A TRUST REGION ALGORITHM VIA BILEVEL LINEAR PROGRAMMING FOR SOLVING THE GENERAL MULTICOMMODITY MINIMAL COST FLOW PROBLEMS

This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems. Using the duality theory of the linear programming and convex theory, the generalized directional derivative of the general multicommodity minimal cost flow problems is derived. The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.     

Scatter Search for Network Design Problem

A fixed charge capacitated multicommodity network design problem on undirected networks is addressed. At the present time, there exists no algorithm that can solve large instances, common in several applications, in a reasonable period of time. This paper presents an efficient procedure using a scatter search framework. Computational experiments on a large set of randomly generated problems show that this procedure is capable of finding good solutions to large-scale problems within a reasonable amount of time.     

Dynamic control of multicommodity fleet management problems

Dynamic fleet management problems with multiple equipment types and limited substitution can be modeled as dynamic, multicommodity network flow problems. These problems are further complicated by the presence of time windows on task arcs (a task, or load, can be handled at different points in time) and the need for integer solutions. In this paper, we formulate the problem as a dynamic control problem, and show that we can produce solutions within four to five percent of a linear relaxation. In addition, we can solve the ultra-large problems that arise in certain applications; these problems are beyond the capabilities of state-of-the-art linear programming solvers.     

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.     

An approximate max-flow min-cut relation for undirected multicommodity flow,with applications

In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF clauses with an arbitrary number of literals per clause.Most of the results in this paper were presented in preliminary form in Approximation through multicommodity flow,Proceedings, 31th Annual Symposium on Foundations of Computer Science (1990), pp. 726–737.Research supported by the National Science Foundation under NSF grant CDA 8722809, by the Office of Naval and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146, and ARPA Order No. 6320, Amendament 1.Research supported by NSF grant CCR-9012357 and by an NSF Presidential Young Investigator Award.