The Multicommodity Flow Problem Updated 21 April 2008

Problem Inputs Multicommodity Flows Slide 2

LP Formulation Multicommodity Flows Slide 3

Multicommodity Flows Figure 17.3 from AMO (costs for all k) Slide 4

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

Multicommodity Flows Figure from AMO (Commodities) CommoditySourceSinkUnits Slide 6

Multicommodity Flows Routing for Commodities 1, 2, and Slide 7

Multicommodity Flows Routing for Commodity Slide 8

Multicommodity Flows Total Flow Slide 9

Multicommodity Flows Example kstb Slide 10

Multicommodity Flows Example 2: Routing for Commodity Cost = 0.5 kstb Slide 11

Multicommodity Flows Example 2: Routing for Commodity Cost = 0.5 kstb Slide 12

Multicommodity Flows Example 2: Routing for Commodity Cost = 0.5 kstb Slide 13

Multicommodity Flows Example 2: Total Flow Cost = kstb Slide 14

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

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

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