1. EMIS 8374 Arc-Path Formulation Updated 2 April 2008

2. 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

3. Node-Arc Formulation

4. 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

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
(4,5)
(2,4) 1
(2,4) 6 t s
(5,6)
(7,7)
(5,5) 3 5
f1 = 2

9. 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

10. 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

11. 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