1. The Multicommodity Flow Problem Updated 21 April 2008

2. Problem Inputs Multicommodity Flows Slide 2

3. LP Formulation Multicommodity Flows Slide 3

4. Multicommodity Flows Figure 17.3 from AMO (costs for all k) 1423 20 5867 10 5 555 9121011 5 5 5 13161415 00 0 5 555 5555 Slide 4

5. Multicommodity Flows Figure 17.3 from AMO (U ij ) 1423  15  5867     9121011  15  13161415      Slide 5

6. Multicommodity Flows Figure 17.13 from AMO (Commodities) CommoditySourceSinkUnits 11410 258 391210 4131610 Slide 6

7. Multicommodity Flows Routing for Commodities 1, 2, and 4 1423 5867 9121011 13161415 10 Slide 7

8. Multicommodity Flows Routing for Commodity 3 1423 5867 9121011 13161415 555 555 5 5 Slide 8

9. Multicommodity Flows Total Flow 1423 5867 9121011 13161415 10 15 10 15 55 Slide 9

10. Multicommodity Flows Example 2 2 13 kstb 1321 2131 3211 Slide 10

11. Multicommodity Flows Example 2: Routing for Commodity 1 2 13 0.5 Cost = 0.5 kstb 1321 2131 3211 Slide 11

12. Multicommodity Flows Example 2: Routing for Commodity 2 2 13 0.5 Cost = 0.5 kstb 1321 2131 3211 Slide 12

13. Multicommodity Flows Example 2: Routing for Commodity 3 2 13 0.5 Cost = 0.5 kstb 1321 2131 3211 Slide 13

14. Multicommodity Flows Example 2: Total Flow 2 13 1 1 0.5 Cost = 1.5 1 0.5 kstb 1321 2131 3211 Slide 14

15. Multicommodity Flows Example 2: Optimal Integral Flow 2 13 Cost = 2 1 (k =1) 1 (k = 2) 1 (k = 3) kstb 1321 2131 3211 Slide 15

16. Multicommodity Flows 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. Slide 16

17. Multicommodity Flows 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. Slide 17