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The Multicommodity Flow Problem

The Multicommodity Flow Problem. Updated 21 April 2008. Problem Inputs. LP Formulation. 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.

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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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