1. Assignment Problem Example
Matrix of costs X Y Z A 1 5 6 B 7 11 C 3 4 8

2. Set up initial residual graph
# – edge costs A 1 X 5 6 5 s 7 B Y t 11 3 4 8 C Z

3. Add in initial prices # – edge costs # – prices A X s B Y t C Z1 X 5 1 4 6
Minimum of
incoming edges
zero’s every place else 6 5 s 7 B Y t 11 3 4 8 C Z

4. Calculate reduced edge costs: cpu,v = cu,v + p(u)-p(v)
# – prices
# – reduced edge costs A 1 X 1 5 1 1 6 4 5 4 3 s 7 B Y t 11 5 4 2 6 3 4 8 2 C Z 6

5. Calculate shortest paths using Dijkstra’s using reduced costs
# – edge costs
# – prices
# – reduced edge costs 8 3 1 5 6 7 11 4 A X 1 1 1 4 4 3 s B Y t 5 4 2 6 2 C Z 6

6. Calculate shortest paths using Dijkstra’s using reduced costs
# – prices
# – reduced edge costs A X 1 1 1 4 4 3 s B Y t 5 4 2 6 2 C Z 6

7. Calculate shortest paths using Dijkstra’s using reduced costs
# – prices
# – reduced edge costs A X 1 1 1 4 4 3 s B Y t 5 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 2 6 2 C Z 6

8. Shortest path to t is augmenting path
# – prices
# – reduced edge costs A X 1 1 1
Min path to t 4 4 3 s B Y t 5 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 2 6 2 C Z 6

9. Use it to get Minimum Cost Matching of size 1
# – edge costs
# – prices
# – reduced edge costs 1 A X 1 1 1 4 4 3 s B Y t 5 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 2 6
Min cost matching, size 1
A -> X cost = 1 2 C Z 6

10. Rebuild graph using minimum cost matching
# – edge costs
# – old prices A -1 X 5 1 6 5 s 7 B Y t 11 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 3 4
Min cost matching, size 1
A -> X cost = 1 8 C Z 6

11. Calculate new prices: p(v) = p(v) + dps,v
# – edge costs
# – old prices A -1 X 5 1 6 5 s 7 B Y t 11 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 3 4
Min cost matching, size 1
A -> X cost = 1 8 C Z 6

12. Calculate new prices: p(v) = p(v) + dps,v (in this case, no change)
# – edge costs
# – prices A -1 X 5 1 6 5 s 7 B Y t 11 4
Distance dps,v
A: 0
B: 0
C: 0
X: 0
Y: 0
Z: 0 3 4
Min cost matching, size 1
A -> X cost = 1 8 C Z 6

13. Calculate reduced edge costs: cpu,v = cu,v + p(u)-p(v)
(note: all reduce costs ≥ 0)
# – edge costs
# – prices
# – reduced edge costs -1 A X 5 1 1 6 4 5 4 3 s 7 B Y t 11 5 4 2 6 3 4
Min cost matching, size 1
A -> X cost = 1 8 2 C Z 6

14. Calculate shortest paths using Dijkstra’s using reduced costs
# – prices
# – reduced edge costs A X 1 1 4 4 3 s B Y t 5 4 2 6
Min cost matching, size 1
A -> X cost = 1 2 C Z 6

15. Calculate shortest paths using Dijkstra’s using reduced costs
# – prices
# – reduced edge costs A X 1 1 4 4 3 s B Y t 5 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 2 6
Min cost matching, size 1
A -> X cost = 1 2 C Z 6

16. Shortest path to t is next augmenting path
# – prices
# – reduced edge costs A X 1 1 4 4 3 s B Y t 5 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 2 6
Min cost matching, size 1
A -> X cost = 1 2 C Z 6

17. Use it to get Minimum Cost Matching of size 2
# – edge costs
# – prices
# – reduced edge costs A X 1 1 4 4 3 s B Y t 5 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 2 6 4
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 2 C Z 6

18. Rebuild graph using minimum cost matching
# – edge costs
# – old prices A -1 X 5 1 6 5 s 7 B Y t 11 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 3 -4
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 8 C Z 6

19. Calculate new prices: p(v) = p(v) + dps,v
# – edge costs
# – old prices A -1 X 5 1 6 5 s 7 B Y t 11 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 3 -4
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 8 C Z 6

20. Calculate new prices: p(v) = p(v) + dps,v
# – edge costs
# – prices A -1 X 5 2 3 6
changed 5 s 7 B Y t 11 4
Distance dps,v
A: 2
B: 0
C: 0
X: 2
Y: 0
Z: 2 3 -4
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 8 C Z 8
changed

21. Calculate reduced edge costs: cpu,v = cu,v + p(u)-p(v)
# – prices
# – reduced edge costs -1 A X 5 3 2 3 6 2 5 3 s 7 B Y t 11 3 4 8 3 -4
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 8 C Z 8

22. Calculate shortest paths using Dijkstra’s using reduced costs
# – edge costs
# – prices
# – reduced edge costs A X 3 2 3 2 3 s B Y t 3 4 8
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 C Z 8

23. Calculate shortest paths using Dijkstra’s using reduced costs
# – edge costs
# – prices
# – reduced edge costs A X 3 2 3 2 3 s B Y t 3 4
Distance dps,v
A: 2
B: 0
C: 3
X: 2
Y: 3
Z: 2 8
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 C Z 8

24. Shortest path to t is next augmenting path
# – edge costs
# – prices
# – reduced edge costs A X 3 2 3 2 3 s B Y t 3 4
Distance dps,v
A: 2
B: 0
C: 3
X: 2
Y: 3
Z: 2 8
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 C Z 8

25. Use it to get Minimum Cost Matching of size 3
# – edge costs
# – prices
# – reduced edge costs -1 A X 3 2 3 6 2 5 3 s B Y t 3 4
Distance dps,v
A: 2
B: 0
C: 3
X: 2
Y: 3
Z: 2 8
Min cost matching, size 2
A -> X cost = 1
C -> Y cost = 4 C Z 8

26. Use it to get Minimum Cost Matching of size 3
# – edge costs
# – prices
Undo previous matching
# – reduced edge costs -1 A X 3 2 3 6 2 5 3 s B Y t 3 4
Distance dps,v
A: 2
B: 0
C: 3
X: 2
Y: 3
Z: 2 8
Min cost matching
C -> Y cost = 4 C Z 8

27. Use it to get Minimum Cost Matching of size 3
# – edge costs
# – prices
# – reduced edge costs A -1 X 3 2 3 6
Add two new matchings 2 5 3 s B Y t 3 4
Distance dps,v
A: 2
B: 0
C: 3
X: 2
Y: 3
Z: 2 8
Min cost matching, size 2
A -> Z cost = 6
C -> Y cost = 4
B -> X cost = 5 C Z 8

28. Final matching, cost = 15 A X 6 5 B Y 4 C Z