Arc-Path Formulation: Example Max Flow Problem

1. EMIS 8374Arc-Path FormulationUpdated 2 April 2008

2. Example Max Flow Problem (0,2) 2 4 (0,4) (0,5) 1 6 (0,4) t s (0,6) (0,7) (0,5) 3 5

3. Node-Arc Formulation

4. Arc-Path Formulation: Notation • Notation: • P denotes the set of paths from s to t • Ap denotes the set of arcs in path p • Uij denotes the set of paths that use arc (i, j) • uij denotes the capacity of arc (i, j) • Decision variables • fp denotes the amount of flow on path p • v denotes the amount of flow sent from s to t

5. Arc-Path Formulation: LP

6. Arc-Path Formulation for Example Problem • Pst = {1, 2, 3} where • A1={(1, 2), (2, 4), (4, 6)} • A2={(1, 2), (2, 5), (5, 6)} • A3= {(1, 3), (3, 5), (5, 6)} • U1,2 = {1, 2}, U1,3 = {3} • U2,4 = {1}, U2,5 = {2} • U3,5 = {3}, U4,6 = {1}, U5,6 = {2, 3}

7. Arc-Path Formulation: LP

8. Example Max Flow Solution (2,2) 2 4 (2,4) (4,5) 1 6 (2,4) t s (5,6) (7,7) (5,5) 3 5 f1 = 2

9. Example Max Flow Solution (2,2) 2 4 (2,4) (4,5) 1 6 (2,4) t s (5,6) (7,7) (5,5) 3 5 f1 = 2 f2 = 2

10. Example Max Flow Solution (2,2) 2 4 (2,4) (4,5) 1 6 (2,4) t s (5,6) (7,7) (5,5) 3 5 f1 = 2 f2 = 2 f3 = 5

11. Arc-Path Model: Pros and Cons • Pros • Facilitates modeling side constraints on paths (e.g., hop-count limits) • Explicitly maps flows to paths • Cons • Doesn’t guarantee integer solutions • Requires explicitly listing paths in P