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IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007

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. Contents. 1. Basic definitions.

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IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007

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

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

  4. Example v2 v5 M v3 v4 v1 Cont(G,M) v6 v7 Basic definitions • Consider G=(V,E) a non-oriented graph and MV. Definition:v is controlled by MV  |NG[v]M||NG[v]|/2

  5. M 0 1 2 3 4 5 Basic definitions • Cont(G,M) → set of vertices controlled by M. • M defines a monopoly in GCont(G,M) = V. Given G=(V,E) and MV:

  6. 0 1 2 0 1 2 3 4 5 3 4 5 G1=(V,E1) G2=(V,E2) Basic definitions • Sandwich Graph 0 1 2 3 4 5 G=(V,E) where E1E  E2

  7. Basic definitions • Monopoly Verification Problem – MVP • Given G1(V,E1), G2(V,E2) and MV, G=(V,E) s.t. E1 E  E2 and M is monopoly in G ? • Solved in polynomial time (Makino, Yamashita, Kameda, Algorithmica [2002]).

  8. Basic definitions - Max-Controlled Set Problem – 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. • The MCSP is NP-hard !! (Makino et al.[2002]).

  9. Basic definitions • MCSP M 2 1 0 5 3 4 6 Not-controlled vertices Fixed Edges Optional Edges Controlled vertices

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

  11. (1) (0) (4) M 0 1 2 Vertices not-controlled by M 3 4 5 (4) (3) (-2) -controlled vertices by M GMCSP • f-controlled vertices • A vertex iV is -controlled by MV iff, |NG[i]M|-|NG[i]U| i , withi Z and U=V \ M. f i fixed gaps (for i V)

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

  13. GMCSP • Generalized Max-Controlled Set Problem • INPUT: Given G1(V,E1), G2(V,E2) and MV (with fixed gaps and positive weights). • OBJECTIVE: 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. GMCSP • Reduction Rules: U=V\M M We delete all optional edges We fix all optional edges

  15. (0)[1] (0)[1] (0)[1] M 0 1 2 3 4 5 (0)[1] (0)[1] (0)[1] GMCSP • Reduction Rules E1D(M,M)  E  E1D(M,M)D(U,M) Vertices not-controlled Fixed Edges -controlled vertices Optional Edges

  16. GMCSP • Reduction Rules • Consider the following partition of V: • MAC and UAC  vertices always -controlled • MNC and UNC _ vertices never -controlled • MR and UR  vertices -controlled or not.

  17. UAC UR UNC MAC MR MNC M U GMCSP • Reduction Rules

  18. UAC UR UNC MAC MR MNC M U GMCSP • Reduction Rules fixed edges optional edges

  19. PMCCG • Reduction Rules (0)[1] (0)[1] (0)[1] M 0 1 2 MSC={1} UNC={5} 3 4 5 (0)[1] (0)[1] (0)[1] Vertices not-controlled by M Fixed Edges -controlled vertices by M Optional Edges

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

  21. GMCSP • ½-Approximation algorithm - GMCSP • Algorithm 1 1: W1 Summation of all weights for E=E1 2: W2 Summation of all weights for E=E2 3: zH1  max{W1,W2}

  22. (0)[5] (0)[1] (0)[3] M 0 1 2 3 4 5 (0)[2] (0)[1] (0)[3] GMCSP • ½-approximation for the GMCSP W1=9 W2=7 Not -controlled vertices Fixed Edges Optional Edges f-controlled vertices

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

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

  25. (2) M GMCSP • Consider M (1) bi=3 bi=3

  26. PMCCG • Stronger LP Formulation Subject to:

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

  28. GMCSP Theorem: Consider a relaxed solution of with . and . If for some (i,j)E2, then there exists another relaxed solution with and

  29. M 0 1 2 0,5 0,5 0,5 0,5 3 4 PMCCG • Feasible solution based in the Linear Relaxation M 0 1 2 1 0 1 0 3 4 Not-controlled vertices Fixed edges Optional edges Controlled vertices

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

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

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

  33. MCSP • Combined Heuristic- CH • 1) z1 ½-approximation • 2) z2 Based LP Heuristic • 3) z  max{z1 , z2} (Martinhon&Protti, LNCC[2002]) MCSP  Similar combined heuristic with ratio:

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

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

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

  37. THANK YOU !!

  38. UAC UR UNC MAC MR MNC M U GMCSP • Reduction Rules • Rule 3: Add to E1 all edges of D(MACMNC, UR). • Rule 4: Remove from E2 the edges D(MR,UACUNC). • Rule 5: Add or remove at random the edges D(MACMNC, UACUNC).

  39. GMCSP • Reduction Rules • Given two graphs G1 e G2, and 2 subsets A,BV, we define: D(A,B)={(i,j)E2\E1 | iA, jB} • Rule 1: Add to E1 the edges D(M,M). • Rule 2: Remove from E2 the edges D(U,U).

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