1. Design & Analysis of Algorithms Combinatorial optimization
SCHOOL OF COMPUTING
Design & Analysis of Algorithms Combinatorial optimization
Pasi Fränti

2. Optimization techniques
Local search
Stochastic variations of local search
Genetic algorithms
Swarm Intelligence

3. Knapsack problem Simplified
Input: Weight of N items {w1, w2, ..., wn}
Knapsack limit S
Output: Selection for knapsack: {x1,x2,…xn}
where xi {0,1}. 5 3 7
Sample input:
wi={2,3,5,7,11}
S=15
Max 15 2 11

4. Knapsack as decision tree
Yes No
Chooce 2 2 -
Yes No
Yes No
Chooce 3 5 2 3 -
Yes No
Yes No
Yes No
Yes No
Chooce 5 10 5 7 2 8 3 5 -
Yes No
Yes
Chooce 7 17 10 12 5 5 14 9 2 15 8 10 3 12 5 7 - … … … … No 28 12 14 13 26 15 12 -

5. In context of search tree
Brute force: search full tree 2 - 5 2 3 -
Heuristics
Greedy 10 5 7 2 8 3 5 - 17 10 12 5 5 14 9 2 15 8 10 3 12 5 7 - … … … 28 10 12 14 13 26 15 12 -
Local search

6. Local search

7. Main principle of local search
Select one and move

8. Structure of local search
Generate initial solution
REPEAT
Generate a set of new solutions
Evaluate the new solutions
Select the best solution
UNTIL stopping criterion met
best 13 15 9 12 11 7 13 21 9 12 14 17
best

9. Components of local search
Representation of solution
Bit string ( …)
Problem-domain data structure {1, 3, 5, 2…}
Neighborhood function (move)
Bits: ( …)  ( …)
Application domain: {1, 3, 5, 2…}  {1, 5, 3, 2…}
Search strategy
Best-improvement: try all, select best
First-improvement: select first that improves.
TSP

10. Movement in neighborhood
Study neighbor solutions

11. Search strategy Best improvement: First improvement: first best13 21 9 12 14 17 12
first
best 21 13
Hill-climbing (max!) /
Descendent (min! 15 13 21 9 12 14 17 8
dead end 7 6 8 11 10 9 5 4 8 7

12. Accept only better solutions
Hill climbing
Accept only better solutions

13. Local and global maxima

14. Combining local search and hill-climbing

15. Local search for Knapsack

16. Representation of knapsack
Represent solution as bit string (x1x2,…xn), where xi {0,1}.
Problem instance: wi= (2,3,5,7,11), S=15.
Solution with elements 2,3 and 7 is represented as

17. Move in knapsack
S=15 W=[2, 3, 5, 7, 11]
Single bit change: 01 or 10

18. Extended neighborhood
S=15 W=[2, 3, 5, 7, 11]
Two operations:
01 or 10
Swap bit location
10011

19. Local maximum
S=15 W=[2, 3, 5, 7, 11]

20. Tabu search
S=15 W=[2, 3, 5, 7, 11]
Prevents search to return previously visited solutions
Tabu!
Select the next best

21. Tabu search (2nd iteration)
S=15 W=[2, 3, 5, 7, 11]

22. Traveling salesman problem
Permute local changes in given route
...  pi-1  pi  pi+1  ...
...  pi-1  pi+1  pi  ...
...  pi  pi-1  pi+1  ...
...  pi  pi+1  pi-1  ...
...  pi+1  pi  pi-1  ...
...  pi+1  pi-1  pi  ...

23. Local search for TSP

24. TSP example E  F  G  H  A 4 +  + 2 +  = 2 + 6
E  F  H  G  A = min!
E  G  F  H  A  +   = 3 + 3
E  G  H  F  A   = 2 + 5
E  H  G  F  A  +  = 2 + 5
E  H  F  G  A  + 2 = 1 + 6

25. Genetic algorithm

26. Genetic algorithm (GA)
Needs more material!

27. Main structure of GA Generate a set of initial solutions. REPEAT
Generate new solutions by crossover.
Mutate the new solutions (optional).
Evaluate the candidate solutions.
Retain best candidates and delete the rest.
UNTIL stopping criterion met.

28. Permuting pairs for crossover
Elitist approach using
zigzag scanning among the best solutions

29. Chess playing

30. Minmax technique

31. Tic-tac-toe example

32. Optimizing chess playing
Revise

33. Evaluation function

34. Minmax example
Redraw 34

35. Minmax playing: Min’s move35

36. Minmax playing: Max’s move
Minimax maximizes the
worst-case outcome for max 36

37. Chess Game tree

38. Beyond the horizon
What value is the position?

39. Evaluating Chess position

40. Positional factors

41. Initial value range

42. Result of optimization

43. Swarm intelligence

44. Swarm intelligence (SI)
Social intelligence: individual behavior maybe naive but joint effect can be intelligent.
Decentralized: no central control of the individuals of the colony
Self-organized: individual adapts to environment and other members of colony
Robust: Task is completed even if some individuals fail

45. Ant colony optimization (ACO)
Main principle:
Emitting pheromone between nest and food
Joint efforts to carry loads
Solving TSP by ants:
Sending ants to make randomized tours
Short links chosen more often than long ones
Good tracks are marked by pheromone
Tracks with high pheromone chosen more often

46. Ant colony optimization
Initialization:
Generate randomized tours.
Smaller links chosen more often than longer ones
Simulate pheromone:
Subtract cost of the links in best solution (-1)
Increase the ones in the worst solution (+1)
Tours are evaluated using the original graph.

47. Ant colony optimization 1st round

48. Ant colony optimization 2nd round

49. The end