EMIS 8374 Arc-Path Formulation Updated 2 April 2008

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

Node-Arc Formulation

Arc-Path Formulation: 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

Arc-Path Formulation: LP

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}

Arc-Path Formulation: LP

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

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

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

Arc-Path Model: Pros and Cons 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