1. 1 COMBINATORIAL OPTIMIZATION : an instance s : Solutions Set f : s → Cost function to minimize (Max) Find s* S s.t. f ( s* ) f ( s ), s S ( MIN) or f ( s* ) f ( s ), s S ( MAX)

2. 2 Local Search ( LS) Neighborhood structure : - i solution - N : S → i→ N ( i ) S N ( i ) = ¨ near ¨ to i solutions ĩ is a local minimum if f ( i ) f ( j ), j N ( i ) ĩ is a local maximum if f ( i ) f ( j ), j N ( i )

3. 3 Local Search Algorithm Define a neighborhood N ( ) Initial solution = Find a solution ΄ N ( ) improving the cost : = ´ If ´ does not exist STOP ( local optimum)

4. 4 The local search algorithm

5. 5 Examples The Traveling Salesman Problem 2 - opt, 3 – opt,..., k – opt 2 - exchange

6. 6

7. 7 Examples The Bipartitioning of a weighted graph G ( V, E, W ), = 2 n. Find partitions A, B of V with = and Minimizing f ( A, B ) =

8. 8 Graph Bipartitioning

9. 9 Search strategies in LS First improvement Best improvement Worst improvement

10. 10 The Quadratic Assignment Problem (QAP) n locations : distance n facilities: flow f ij π(i)=k: facility i  location k minimize the local cost

11. 11 The Quadratic Assignment Problem (QAP) n locations : distance n facilities: flow f ij π(i)=k: facility i  location k minimize the total cost

12. 12 The QAP  2-exchange: π=(π(1),...,π(i),...,π(j),...π(n)) π ij =(π(1),...,π(j),...,π(i),...π(n)) N(π)=(n*(n-1))/2

13. 13 The QAP: an example π=3 1 2 5 4 6 2-exch. π ij =3 4 2 5 1 6 6 5 1 3 4 2 6 5 1 4 3 2 locations

14. 14 Traveling Salesman Problem (2-exchange)

15. 15 Bipartitioning weighted graph G(V,E) 2-exchange

16. 16 Particular cases(Bipartitioning) h/2

17. 17 K-densest and k-lightest

18. 18 Results (2-exchange) m n-m

19. 19 The Local Search: The MIS example : The maximum Independent Set problem in a graph G(V,E)

20. 20 The MIS by the Local Search Solution coding : Function :

21. 21 Neighborhood : FLIP

22. 22 LS Drawbacks Local optimum “good“ neighborhoods exploration strategies Performances guarantee ? Parallelization ?