A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem

Farvolden, Judith M.; Powell, Warren B.; Lustig, Irvin J.

doi:10.1287/opre.41.4.669

Cited by 62 publications

(35 citation statements)

References 24 publications

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“…Moreover, it would be interesting to compare the specialized GUB approach developed in our Dantzig-Wolfe algorithm with the partitioning technique recently proposed by Farvolden et al (1993) to see if improvements can be achieved. Further research also includes a better theoretical understanding of the results produced, the development of decomposition algorithms based on right-hand-side allocation, and the application of decomposition techniques to other types of multicommodity flow problems.…”

Section: Resultsmentioning

confidence: 99%

“…First, it is not always a good idea to replace a sparse system with a more compact formulation. Indeed, as Farvolden et al (1993), we speculate that the number of extreme points enumerated depends on whether the master program is finding a convex combination of path flows or tree flows and that if the extreme points are path flows, fewer extreme points will be enumerated. Secondly, convexity constraints in both methods can be handled very Table 3.…”

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confidence: 96%

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Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems

Chardaire

Lisser

2002

Operations Research

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Multicommodity network flow models arise in a wide variety of contexts, typical among which is the dimensioning of telecommunication networks. In this paper, we present various approaches based on specialization of the simplex algorithm and interior-point methods to solve nonoriented multicommodity flowproblems. Algorithms are tested with data from the France-Telecom Paris district transmission network. First, we focus on a specialization for the node-arc formulation of the problem. A Primal simplex and Dual Affine Scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared. Numerical results are provided for problems up to about 800,000 constraints and 6,000,000 variables. However, much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem. A Dantzig-Wolfe decomposition method is designed and compared with a specialized implementation of the Analytic Center Cutting Plane Method (ACCPM). Partitioning techniques are used to exploit the structure of the master programs involved in those methods.

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Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 96%

Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems

Chardaire

Lisser

2002

Operations Research

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“…This simplex approach is commonly known as the network simplex algorithm (Ahuja, et al, 1993:665). These matrices are triangular, enabling the system of linear computation for arc-chain formulation (Ford and Fulkerson, 1958;Farvolden, Powell and Lustig, 1993;Goffin, Gondzio, Sarkissian and Vial, 1995:1).…”

Section: Multicommodity Network Flowmentioning

confidence: 99%

Modeling and Analysis of Multicommodity Network Flows Via Goal Programming

Scott¹,

Deckro²,

Chrissis³

2005

INFOR: Information Systems and Operational Research

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In this work goal programming is used to solve a minimum cost multicommodity network flow problem with multiple goals. A single telecommunication network with multiple commodities (e.g., voice, video, data, etc.) flowing over it is analyzed. This network consists of: linear objective function, linear cost arcs, fixed capacities, specific origin-destination pairs for each commodity. A multicommodity network flow problem with goals can be successfully modeled using linear goal programming techniques. When properly modeled, network flow techniques may be employed to exploit the pure network structure of a multicommodity network flow problem with goals. Lagrangian relaxation captures the essence of the pure network flow problem as a master problem and sub-problems (McGinnis and Rao, 1977). A subgradient algorithm may optimize the Lagrangian function, or the Lagrangian relaxation could be decomposed into subproblems per commodity; each subproblem being a single commodity network flow problem. Parallel to the decomposition of the Lagrangian relaxation, Dantzig-Wolfe decomposition may be implemented to the linear program. Post-optimality analyses provide a variety of options to analyze the robustness of the optimal solution. The options of post-optimality analysis consist of sensitivity analysis and parametric analysis. This mix of modeling options and analyses provide a powerful method to produce insight into the modeling of a multicommodity network flow problem with multiple objectives. Subject TermsMulticommodity Network Flow, Goal Programming, Linear Programming, Lagrangian Relaxation, Decomposition Report Classification unclassified Classification of this page unclassified Classification of Abstract unclassified Limitation of Abstract UU Number of Pages 115The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U. S. Government. AFIT/GOR/ENS/02-17 MODELING AND ANALYSIS OF MULTICOMMODITY NETWORK FLOWS VIA GOAL PROGRAMMING THESIS AbstractInformation superiority, along with air superiority, must be achieved and maintained in a theatre of battle in order to increase efficacy and provide protection to our forces, both on the ground and in the air. Informa tion, for the most part, utilizes an underlying network that is identified, and is of vital interest to a commander in a wartime or peacetime scenario. This underlying network must be modeled and analyzed in order

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“…The reason for this at first surprising fact is that large multicommodity flow instances have to be solved in many application areas and in combination with a whole variety of discrete optimization problems such as network design or graph bisection. And standard simplex or interior point LP-solvers, even specialized software based on primal basis partitioning [4,5] or Lagrangian relaxation based resource-or cost-decompositions [3,9] are simply not fast enough to tackle real-size instances of these problems very efficiently. Therefore, there is a big interest in the development of algorithms that provide near optimal solutions more quickly.…”

Section: The Lp-formulationmentioning

confidence: 99%

Multicommodity Flow Approximation Used for Exact Graph Partitioning

Sellmann

Sensen

Timajev

2003

Algorithms - ESA 2003

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Abstract. We present a fully polynomial-time approximation scheme for a multicommodity flow problem that yields lower bounds of the graph bisection problem. We compare the approximation algorithm with Lagrangian relaxation based cost-decomposition approaches and linear programming software when embedded in an exact branch&bound approach for graph bisection. It is shown that the approximation algorithm is clearly superior in this context. Furthermore, we present a new practical addition to the approximation algorithm which improves its performance distinctly. Finally, we prove the performance of the graph bisection algorithm using multicommodity flow approximation by computing formerly unknown bisection widths of some DeBruijn-and Shuffle-Exchange-Graphs.

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A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem

Cited by 62 publications

References 24 publications

Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems

Simplex and Interior Point Specialized Algorithms for Solving Nonoriented Multicommodity Flow Problems

Modeling and Analysis of Multicommodity Network Flows Via Goal Programming

Multicommodity Flow Approximation Used for Exact Graph Partitioning

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