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The Multicommodity Flow Problem

The multicommodity flow problem (MCFP) presents significant challenges due to its inherent complexity, classified as NP-hard. This paper discusses the LP formulation of the MCFP and details illustrations from "Algorithms for Optimization" (AMO). It emphasizes the intricacies introduced by bundling constraints, which complicate the problem compared to standard network flow issues. The MCFP is vital in optimizing routing for multiple commodities simultaneously and has no known polynomial-time solutions, making it a critical area of study in theoretical computer science and operations research.

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The Multicommodity Flow Problem

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  1. The Multicommodity Flow Problem Updated 21 April 2008

  2. Problem Inputs

  3. LP Formulation

  4. Figure 17.3 from AMO (costs for all k) 20 20 20 1 2 3 4 5 5 5 5 10 10 10 5 6 7 8 5 5 5 5 5 5 5 9 10 11 12 5 5 5 5 0 0 0 13 14 15 16

  5. Figure 17.3 from AMO (Uij)  15  1 2 3 4     15   5 6 7 8     15   9 10 11 12      15  13 14 15 16

  6. Figure 17.13 from AMO (Commodities)

  7. Routing for Commodities 1, 2,and 4 1 2 3 4 10 10 10 10 10 5 6 7 8 10 10 10 10 10 9 10 11 12 10 10 10 13 14 15 16

  8. Routing for Commodity 3 1 2 3 4 5 6 7 8 5 5 5 9 10 11 12 5 5 5 5 5 13 14 15 16

  9. Total Flow 1 2 3 4 10 10 10 10 10 5 6 7 8 10 10 15 15 15 9 10 11 12 5 5 15 15 15 13 14 15 16

  10. Example 2 2 1 3

  11. Example 2: Routing for Commodity 1 2 Cost = 0.5 0.5 0.5 1 0.5 3

  12. Example 2: Routing for Commodity 2 2 Cost = 0.5 0.5 0.5 1 3 0.5

  13. Example 2: Routing for Commodity 3 2 Cost = 0.5 0.5 0.5 1 0.5 3

  14. Example 2: Total Flow 2 Cost = 1.5 0.5 0.5 1 1 1 1 3 0.5

  15. Example 2: Optimal Integral Flow 2 Cost = 2 1 (k =1) 1 (k = 3) 1 (k = 3) 1 3 1 (k = 2)

  16. Complexity • The bundling constraints make the multicommodity flow problem with integral flows significantly more difficult to solve than pure network flow problems. • This problem belongs to the class of theoretically intractable NP-hard optimization problems.

  17. NP-hard Problems • Multicommodity Flow belongs to the class of NP-hard problems for which no known polynomial time algorithms exist. • Other NP-hard problems: TSP, network design, longest path, knapsack, integer programming. • If there exists a polynomial time algorithm for any NP-hard problem, then there is one for every NP-hard problem. • Whether or not such an algorithm exists is a fundamental unsolved problem in theoretical computer science and OR.

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