A Difficult Graph for Independence Number Theory

Transcription

1 A Difficult Graph for Independence Number Theory Craig Larson (joint work with Patrick Gaskill) Virginia Commonwealth University Richmond, VA CanaDAM June 13, 2013

2 The Independence Number of a Graph

3 The Independence Number of a Graph The independence number α of a graph is the largest number of mutually non-adjacent vertices.

4 The Independence Number of a Graph The independence number α of a graph is the largest number of mutually non-adjacent vertices. α = 4.

5 The Main Problem Find an efficient algorithm that computes the independence number for this graph.

6 Some Facts

7 Some Facts Order n = 11.

8 Some Facts Order n = 11. Size e = 23.

9 Some Facts Order n = 11. Size e = 23. Minimum Degree δ = 3

10 Some Facts Order n = 11. Size e = 23. Minimum Degree δ = 3 Maximum Degree = 5.

11 Good Efficiently Computable Upper Bounds for α The Lovász number of a graph G is: ϑ(g) = max[1 λ 1(A) λ n (A) ] over all real matrices A with a ij = 0 if v i v j in G, with eigenvalues λ 1 (A)... λ n (A)

12 Good Efficiently Computable Upper Bounds for α The Lovász number of a graph G is: ϑ(g) = max[1 λ 1(A) λ n (A) ] over all real matrices A with a ij = 0 if v i v j in G, with eigenvalues λ 1 (A)... λ n (A) α ϑ = 4.107

13 Good Efficiently Computable Upper Bounds for α The Lovász number of a graph G is: ϑ(g) = max[1 λ 1(A) λ n (A) ] over all real matrices A with a ij = 0 if v i v j in G, with eigenvalues λ 1 (A)... λ n (A) α ϑ = L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory, D. Knuth, The sandwich theorem, Electronic Journal of Combinatorics 1 (1994).

14 Lovász Theta is a very good upper bound for α For all simple graphs,

15 Lovász Theta is a very good upper bound for α For all simple graphs, ϑ predicts α for 34 out of 34 graphs of order 5.

16 Lovász Theta is a very good upper bound for α For all simple graphs, ϑ predicts α for 34 out of 34 graphs of order 5. ϑ predicts α for 156 out of 156 graphs of order 6.

17 Lovász Theta is a very good upper bound for α For all simple graphs, ϑ predicts α for 34 out of 34 graphs of order 5. ϑ predicts α for 156 out of 156 graphs of order 6. ϑ predicts α for 1044 out of 1044 graphs of order 7.

18 Lovász Theta is a very good upper bound for α For all simple graphs, ϑ predicts α for 34 out of 34 graphs of order 5. ϑ predicts α for 156 out of 156 graphs of order 6. ϑ predicts α for 1044 out of 1044 graphs of order 7. ϑ predicts α for out of graphs of order 8.

19 Lovász Theta is a very good upper bound for α For all simple graphs, ϑ predicts α for 34 out of 34 graphs of order 5. ϑ predicts α for 156 out of 156 graphs of order 6. ϑ predicts α for 1044 out of 1044 graphs of order 7. ϑ predicts α for out of graphs of order 8. ϑ predicts α for out of graphs of order 9.

20 A Problem

21 A Problem Characterize Graphs where α = Lovász Theta.

22 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the residue is the number of zeros at the result of the Havel-Hakimi process.

23 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the residue is the number of zeros at the result of the Havel-Hakimi process. 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3.

24 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the residue is the number of zeros at the result of the Havel-Hakimi process. 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3. 0, 0, 0.

25 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the residue is the number of zeros at the result of the Havel-Hakimi process. 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3. 0, 0, 0. R = 3. R α Graffiti, 1988; Favaron, Maheo, Sacle, 1991; Griggs, Kleitman, 1994.

26 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the Caro-Wei bound is

27 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the Caro-Wei bound is 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3.

28 Good Efficiently Computable Lower Bounds for α Given a graph G with degree sequence (d) the Caro-Wei bound is 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3. 1 d(v)+1 = α. Y. Caro, New results on the independence number, Tel-Aviv University technical report (1979) V. Wei, A lower bound on the stability number of a simple graph, Bell Labs Technical Memorandum (1981).

29 Good Efficiently Computable Lower Bounds for α Given a connected graph G the odd minus odd horizontal bound is the maximum, over all vertices v, of n e for the subgraph induced by the vertices at odd distance from v.

30 Good Efficiently Computable Lower Bounds for α Given a connected graph G the odd minus odd horizontal bound is the maximum, over all vertices v, of n e for the subgraph induced by the vertices at odd distance from v. Odd minus odd horizontal = 3. S. Fajtlowicz, Written on the Wall: Conjectures of Graffiti,

31 A Problem Find an efficiently computable lower bound LB α so that LB(J? FBo{fdb?) = 4.

32 A Problem Find an efficiently computable lower bound LB α so that LB(J? FBo{fdb?) = Efficiently Computable Bounds are Known

33 Forbidden Subgraph Characterizations G. Minty, On maximal independent sets of vertices in claw-free graphs, Journal of Combinatorial Theory. Series B, 28 (1980) N. Sbihi, Algorithme de recherche d un stable de cardinalité maximum dans un graphe sans étoile, Discrete Mathematics (1980)

34 Forbidden Subgraph Characterizations M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour, K. Vušković, Recognizing Berge graphs, Combinatorica 25 (2005)

35 Forbidden Subgraph Characterizations A. Hertz, D. de Werra, On the stability of AH-free graphs, Discrete Applied Mathematics (1993)

36 Forbidden Subgraph Characterizations A. Hertz, On the use of Boolean methods for the computation of the stability number, Discrete Applied Mathematics (1997)

37 Forbidden Subgraph Characterizations M. Gerber, A. Hertz, V. Lozin, Stable sets in two subclasses of banner-free graphs, Discrete Applied Mathematics (2004)

38 A Problem

39 A Problem Find a forbidden subgraph characterization that applies to this graph.

40 A Problem Find a forbidden subgraph characterization that applies to this graph. There are at least 15 published forbidden subgraph characterizations.

41 α-reductions

42 α-reductions 1. Is disconnected.

43 α-reductions 1. Is disconnected. 2. Maximum degree = n 1.

44 α-reductions 1. Is disconnected. 2. Maximum degree = n Has twin vertices.

45 α-reductions 1. Is disconnected. 2. Maximum degree = n Has twin vertices. 4. Has a simplicial vertex.

46 α-reductions 1. Is disconnected. 2. Maximum degree = n Has twin vertices. 4. Has a simplicial vertex. 5. Has a non-empty critical independent set (Zhang, 1990).

47 α-reductions 1. Is disconnected. 2. Maximum degree = n Has twin vertices. 4. Has a simplicial vertex. 5. Has a non-empty critical independent set (Zhang, 1990). 6. Has a foldable vertex (Fomin, Grandoni, Kratsch, 2006).

48 α-reductions 1. Is disconnected. 2. Maximum degree = n Has twin vertices. 4. Has a simplicial vertex. 5. Has a non-empty critical independent set (Zhang, 1990). 6. Has a foldable vertex (Fomin, Grandoni, Kratsch, 2006). 7. Has a magnet (Hammer, Hertz, 1991; Hertz, de Werra, 2009).

49 A Problem Find new α-reductions.

50 Separable Independent Sets An independent set I is separable if

51 Separable Independent Sets An independent set I is separable if 1. X = I N(I ),

52 Separable Independent Sets An independent set I is separable if 1. X = I N(I ), 2. X c = V X, and

53 Separable Independent Sets An independent set I is separable if 1. X = I N(I ), 2. X c = V X, and 3. α(g) = I + α(g[x c ]).

54 Separable Independent Sets An independent set I is separable if 1. X = I N(I ), 2. X c = V X, and 3. α(g) = I + α(g[x c ]). Note: critical independent sets are separable independent sets.

55 Separable Independent Sets 1. I =Red, N(I )=Yellow

56 Separable Independent Sets 1. I =Red, N(I )=Yellow 2. X = I N(I ),

57 Separable Independent Sets 1. I =Red, N(I )=Yellow 2. X = I N(I ), 3. X c =Black, and

58 Separable Independent Sets 1. I =Red, N(I )=Yellow 2. X = I N(I ), 3. X c =Black, and 4. 4 = α(g) = I + α(g[x c ]) =

59 Separable Independent Sets 1. I =Red, N(I )=Yellow 2. X = I N(I ), 3. X c =Black, and 4. 4 = α(g) = I + α(g[x c ]) = I is a separable independent set.

60 A Problem

61 A Problem How to Efficiently Identify Separable Independent Sets?

62 A Problem How to Efficiently Identify Separable Independent Sets? What kinds of separable independent sets are there?

63 A Problem How to Efficiently Identify Separable Independent Sets? What kinds of separable independent sets are there? Which kinds can be identified efficiently?

64 The Independence Number Project

65 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory.

66 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex.

67 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if:

68 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5),

69 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or

70 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound.

71 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound. Then we generated all graphs with n = 6,...

72 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound. Then we generated all graphs with n = 6,... Then we generated all graphs with n = 7,...

73 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound. Then we generated all graphs with n = 6,... Then we generated all graphs with n = 7,... Then we generated all graphs with n = 8,...

74 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound. Then we generated all graphs with n = 6,... Then we generated all graphs with n = 7,... Then we generated all graphs with n = 8,... Then we generated all graphs with n = 9,...

75 The Independence Number Project The main idea is to find the smallest graphs whose independence number cannot be efficiently computed (according to existing Independence Number Theory) and use these graphs to help extend the theory. Connected graphs with n 4 either have a degree n 1 vertex, or a foldable vertex. Then we generated all graphs with n = 5, and checked if: they were reducible (and α could be computed in terms of the independence number of a graph with n < 5), they had an α-property, or the best (efficiently computable) upper bound for α equals the best lower bound. Then we generated all graphs with n = 6,... Then we generated all graphs with n = 7,... Then we generated all graphs with n = 8,... Then we generated all graphs with n = 9,... Then we generated all graphs with n = 10,...

76 The Independence Number Project

77 The Independence Number Project But we got stuck at n = 11.

78 The Independence Number Project But we got stuck at n = 11. It demands new theory.

79 Does P=NP?

80 Does P=NP? Bollobás: My hunch is that P=NP, contrary to general belief.

81 Does P=NP? Bollobás: My hunch is that P=NP, contrary to general belief. With greater probability of success: we can extend the class of graphs for which the independence number can be computed efficiently. B. Bollobás, The Future of Graph Theory, Quo Vadis, Graph Theory?, 1993, 5 11.

82 Thank You! The Independence Number Project: independencenumber.wordpress.com