1. Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 0 UPPER =  LOWER = 0

2. Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 2 0 UPPER =  LOWER = 2

3. Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 3 2 0 UPPER =  LOWER = 3

4. Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 4 3 2 0 UPPER = 4 LOWER = 4

5. 34 8 7 12 10 9 3 6 8 2 7 0 34 3 2 0 UPPER = 3 LOWER = 3 Branch & Bound

6. 34 8 7 12 10 9 3 6 8 2 7 0 34 3 6 2 0 UPPER = 3 LOWER = 6 Branch & Bound

7. 34 8 7 12 10 9 3 6 8 2 7 0 UPPER = 3 LOWER = 7 34 3 6 2 7 0 Branch & Bound

8. Tasks are created dynamically Upper bound is shared Lower bound is local Halve set of feasible tours: Explore only those paths that visit city 1 before city 2. Branch & Bound Illustrated on Traveling Salesman Problem

9. 01 32 1 1 1 1 4 4 0:0 01:1 02:4 03:1 14 1 012:2013:5 1 4 0123:3 1 01230:4 1 031:5032:2 41 Partial tour 0, 3, 2 with cost lower bound = 2 LEGEND TSP instance Search Tree Halve the set of feasible tours: Only visit tours where city 1 occurs before city 2.

10. 01 32 1 1 1 1 4 4 0:0 01:1 02:4 03:1 14 1 012:2013:5 1 4 0123:3 1 01230:4 1 031:5032:2 41 Partial tour 0, 3, 2 with cost lower bound = 2 LEGEND upper bound =  TSP instance Search Tree