1. IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007 Puerto Varas - Chile The Generalized Max-Controlled Set Problem Carlos A. Martinhon Fluminense Fed. University Ivairton M. Santos - UFMT Luiz S. Ochi – IC/UFF

2. 2 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

3. 3 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

4. 4 Basic definitions  Consider G=(V,E) a non-oriented graph and M  V. Definition: v is controlled by M  V  |N G [v]  M|  |N G [v]|/2 Example M v1 v2 v3 v4 v5 v6v7 Cont(G,M)

5. 5 Basic definitions Cont(G,M) → set of vertices controlled by M.Cont(G,M) → set of vertices controlled by M. M defines a monopoly in G Cont(G,M) = V.M defines a monopoly in G Cont(G,M) = V. 0 12 345 M Given G=(V,E) and M  V:

6. 6 Basic definitions  Sandwich Graph 012 345 G 1 =(V,E 1 ) 012 345 G=(V,E) where E 1  E  E 2 012 345 G 2 =(V,E 2 )

7. 7 Basic definitions  Monopoly Verification Problem – MVP Given G 1 (V,E 1 ), G 2 (V,E 2 ) and M  V,Given G 1 (V,E 1 ), G 2 (V,E 2 ) and M  V, nopoly in G ?  G=(V,E) s.t. E 1  E  E 2 and M is monopoly in G ? Solved in polynomial time (Makino, Yamashita, Kameda, Algorithmica [2002]).Solved in polynomial time (Makino, Yamashita, Kameda, Algorithmica [2002]).

8. 8 Basic definitions - Max-Controlled Set Problem – MCSP If the answer to the MVP is NO, we have the MCSP!If the answer to the MVP is NO, we have the MCSP! In the MCSP, we hope to maximize the number of vertices controlled by M.In the MCSP, we hope to maximize the number of vertices controlled by M. The MCSP is NP-hard !! (Makino et al.[2002]).The MCSP is NP-hard !! (Makino et al.[2002]).

9. 9 3 Basic definitions  MCSP 0 1 2 5 46 M Fixed Edges Optional Edges Not-controlled vertices Controlled vertices

10. 10 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

11. 11 GMCSP  f-controlled vertices A vertex i  V is  -controlled by M  V iff, |N G [i]  M|-|N G [i]  U|  i, with  i  Z and U=V \ M.A vertex i  V is  -controlled by M  V iff, |N G [i]  M|-|N G [i]  U|  i, with  i  Z and U=V \ M. Vertices not  -controlled by M  -controlled vertices by M 0 12 345 M (0)(4) (1) (3)(-2) (4) f i  fixed gaps (for i  V)

12. 12 GMCSP  We also add positive weights 0 1 4 5 2 3 M (0)[2](0)[3] (0)[5](0)[7](0)[10](0)[1] Fixed Edges Optional Edges Vertices not  -controlled  -controlled vertices

13. 13 GMCSP  Generalized Max-Controlled Set Problem INPUT: Given G 1 (V,E 1 ), G 2 (V,E 2 ) and M  V (with fixed gaps and positive weights).INPUT: Given G 1 (V,E 1 ), G 2 (V,E 2 ) and M  V (with fixed gaps and positive weights). OBJECTIVE: We want to find a sandwich graphOBJECTIVE: We want to find a sandwich graph G=(V,E), in order to maximize the sum of the weights of all vertices f-controlled by M.

14. 14 GMCSP  Reduction Rules: We fix all optional edges We delete all optional edges M U=V\M

15. 15 GMCSP  Reduction Rules 0 12 345 M (0)[1] E 1  D(M,M)  E  E 1  D(M,M)  D(U,M) Fixed Edges Optional Edges Vertices not  -controlled  -controlled vertices

16. 16 GMCSP  Reduction Rules Consider the following partition of V:Consider the following partition of V: –M AC and U AC  vertices always  -controlled –M NC and U NC _  vertices never  -controlled –M R and U R  vertices  -controlled or not.

17. 17 GMCSP  Reduction Rules M AC M R M NC U AC U R U NC MU

18. 18 GMCSP  Reduction Rules M AC M R M NC U AC U R U NC MU optional edges fixed edges

19. 19 PMCCG  Reduction Rules 0 12 345 M (0)[1] M SC ={1} U NC ={5} Fixed Edges Optional Edges Vertices not  -controlled by M  -controlled vertices by M

20. 20 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

21. 21 GMCSP  ½- Approximation algorithm - GMCSP Algorithm 1Algorithm 1 1: W 1  Summation of all weights for E=E 1 2: W 2  Summation of all weights for E=E 2 3: z H1  max{W 1,W 2 }

22. 22 M (0)[5](0)[1](0)[3] (0)[2](0)[1](0)[3] GMCSP  ½-approximation for the GMCSP Not  -controlled vertices f-controlled vertices Fixed Edges Optional Edges 0 12 345 W 1 =9 W 2 =7

23. 23 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

24. 24 GMCSP  LP formulation Consider K=|V|+max{|  i | s.t. i  V}Consider K=|V|+max{|  i | s.t. i  V} Subject to:

25. 25 GMCSP ConsiderConsider M (2) M (1) b i =3

26. 26 PMCCG  Stronger LP Formulation Subject to:

27. 27 Theorem : Let and the optimum values of and respectively. Then: GMCSP Z*=? Optimum objective value What about the feasible solutions? max

28. 28 GMCSP Theorem: Consider a relaxed solution of with. and. If for some (i,j)  E 2, then there exists another relaxed solution with and and

29. 29 PMCCG  Feasible solution based in the Linear Relaxation 012 34 M 0,5 012 34 M 1 0 0 1 Fixed edges Optional edges Not-controlled vertices Controlled vertices

30. 30  Integer solution obtained from our stronger Linear Programming formulation. Algorithm 2Algorithm 2 –Given a relaxed solution for. –Define as  -controlled all vertice i  V with, and not  -controlled if., and not  -controlled if. GMCSP

31. 31 Quality of upper and lower bounds generated by our stronger formulation

32. 32 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

33. 33 MCSP Combined Heuristic - CHCombined Heuristic - CH 1) z 1  ½-approximation1) z 1  ½-approximation 2) z 2  Based LP Heuristic2) z 2  Based LP Heuristic 3) z  max{ z 1, z 2 }3) z  max{ z 1, z 2 } ( Martinhon&Protti, LNCC[2002]) ( Martinhon&Protti, LNCC[2002]) MCSP  Similar combined heuristic with ratio:

34. 34 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

35. 35 Contents 1. Basic definitions 2. The Generalized Max-Controlled Set Problem 3.a) A 1/2-Approximation procedure b) A Based LP Prog. Heuristic b) A Based LP Prog. Heuristic c) A combined heuristic c) A combined heuristic 5. Comp. results and final comments 4. Tabu Search Procedure

36. 36 Computational Results  Tabu Search solutions for instances with 50, 75 and 100 vertices.

37. 37 T HANK Y OU !!

38. 38 GMCSP  Reduction Rules Rule 3: Add to E 1 all edges of D(M AC  M NC, U R ).Rule 3: Add to E 1 all edges of D(M AC  M NC, U R ). Rule 4: Remove from E 2 the edges D(M R,U AC  U NC ).Rule 4: Remove from E 2 the edges D(M R,U AC  U NC ). Rule 5: Add or remove at random the edges D(M AC  M NC, U AC  U NC ).Rule 5: Add or remove at random the edges D(M AC  M NC, U AC  U NC ). M AC M R M NC U AC U R U NC MU

39. 39 GMCSP  Reduction Rules Given two graphs G 1 e G 2, and 2 subsets A,B  V, we define:Given two graphs G 1 e G 2, and 2 subsets A,B  V, we define: D(A,B)={(i,j)  E 2 \E 1 | i  A, j  B} D(A,B)={(i,j)  E 2 \E 1 | i  A, j  B} Rule 1: Add to E 1 the edges D(M,M).Rule 1: Add to E 1 the edges D(M,M). Rule 2: Remove from E 2 the edges D(U,U).Rule 2: Remove from E 2 the edges D(U,U).