tree -decomposition

1. tree -decomposition ＊5499016 竹内　和樹 ＊5499023 藤井　勲

2. contents 1,Introduction 2,Main Algorithm 3,yfiles 4,Construction of Tree-Decomposition 5,Analysis of Algorithm 6,From now on Target

3. Introduction ~Purpose~ INPUT : G = (V,E) Output : Tree-decomposition ( T , X ) １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

4. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I } (1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj

5. Introduction (1), ∪ Xi = V i∈I １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

6. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I } (1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj

7. Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

8. Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

9. Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

10. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I } (1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj

11. Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

12. Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2,4,7 4,6,7 ５

13. Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj １ ２ ６ 1,2,3 2,3,4 4,6,8 ７ ８ ３ ４ 4,5 2, 4,7 4,6,7 ５

14. Main Algorithm ～ An st-separating set ～ a set S ⊆ V {s,t} with the property that any path from s to t 1 2

15. Main Algorithm ～ Minimum st-separating set ～ The minimum number of vertices in an st-separating set

16. Main Algorithm ~Outline~ Input Graph G = ( V , E ) Construct( T , X ) With | I | = 1 and X = V 1 no ∃i∈ I with Gi not a clique Output :current Tree-decomposition yes Calculate minimal separating Vertex set S for Gi: m components Yi1,.....,Yim Define Xi0 : = S Xij : Yij ∪ S, for j = 1,....,m Construct new ( T , X )

17. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５ Input graph G = (V,I)

18. Main Algorithm １ ２ ６ Calculate minimal separataing vertex set S for Gi ７ ８ ３ ４ ５

19. Main Algorithm １ ２ ６ Calculate minimal separataing vertex set S for Gi ７ ８ ３ ４ ５

20. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for Gi . ７ ８ ３ ４ ５ Separate Y and Y S =(4) 11 12 11

21. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for Gi . ７ ８ ３ ４ ５ Separate Y and Y S =(4) 11 12 11

22. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for X . ７ 11 ８ ３ ４

23. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for X . ７ 11 ８ ３ ４

24. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for X . ７ 11 ８ ３ ４ Separate Y and Y S =(2,4) 22 21 21

25. Main Algorithm １ ２ ６ Calculate minimal separating vertex set S for X . ７ 11 ８ ３ ４ Separate Y and Y S =(2,4) 22 21 21

26. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 21 ８ ４

27. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 21 ８ ４

28. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 21 ８ ４ Separate Y and Y S =(4,6) 31 32 31

29. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 21 ８ ４ Separate Y and Y S =(4,6) 31 32 31

30. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 31 ４

31. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 31 ４

32. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 31 ４ Separate Y and Y S =(4,6) 41 42 41

33. Main Algorithm ２ ６ Calculate minimal separating vertex set S for X . ７ 31 ４ Separate Y and Y S =(4,6) 41 42 41

34. Main Algorithm ６ Calculate minimal separating vertex set S for X . ７ 41 ４

35. Main Algorithm ６ Calculate minimal separating vertex set S for X . ７ 41 ４ ・ ・ ・

36. Main Algorithm ６ Calculate minimal separating vertex set S for X . ７ 41 ４ ・ ・ ・ 41 X is clique . r = X = {4,6,7} 1 41

37. Main Algorithm ２ Calculate minimal separating vertex set S for X . ７ 42 ４

38. Main Algorithm ２ Calculate minimal separating vertex set S for Y . ７ 42 ４ ・ ・ ・

39. Main Algorithm ２ Calculate minimal separating vertex set S for Y . ７ 42 ４ ・ ・ ・ 42 X is clique . r = X = {2,4,7} 2 42

40. Main Algorithm ６ Calculate minimal separating vertex set S for X . 32 ８ ４ ・ ・ ・ 32 X is clique . r = X = {4,6,8} 3 32

41. Main Algorithm １ ２ Calculate minimal separating vertex set S for X . 22 ３ ４

42. Main Algorithm １ ２ Calculate minimal separating vertex set S for X . 22 ３ ４

43. Main Algorithm １ ２ Calculate minimal separating vertex set S for X . 22 ３ ４ Separate Y and Y S =(2,3) 33 34 32

44. Main Algorithm １ ２ Calculate minimal separating vertex set S for X . 22 ３ ４ Separate Y and Y S =(2,3) 33 34 32

45. Main Algorithm ２ Calculate minimal separating vertex set S for X . 33 ３ ４ ・ ・ ・ 33 X is clique . r = X = {2,3,4} 4 33

46. Main Algorithm １ ２ Calculate minimal separating vertex set S for X . 34 ３ ・ ・ ・ 34 X is clique . r = X = {1,2,3} 5 34

47. Main Algorithm Calculate minimal separating vertex set S for X . 12 ４ ・ ・ ・ ５ 12 X is clique . r = X = {1,2,3} 6 12

48. Main Algorithm All vertex sets are clique . Connect r ,…, r . 1 6

49. Main Algorithm All vertex sets are clique . S = (2,3) S = (2,4) 32 21 Connect r ,…, r . 1 6 1,2,3 2,3,4 4,6,8 S = (4,6) 31 4,5 2,4,7 4,6,7 S = (4) S = (4,7) 1 41

50. Analysis of algorithm