The canonical augmentation method

Transcription

1 The canonical augmentation method Derrick Stolee 1 University of Nebraska Lincoln s-dstolee1@math.unl.edu s-dstolee1/ May 13, Supported by NSF grants CCF and DMS

2 Generating means...

3 Generating means... Searching (looking for objects)

4 Generating means... Searching (looking for objects) Exhaustively (not missing any)

5 Shifting the Exponent

6 Shifting the Exponent

7 Shifting the Exponent

8 Search by Augmentations While generating [combinatorial object]

9 Search by Augmentations While generating [combinatorial object] we start at [base object]

10 Search by Augmentations While generating [combinatorial object] we start at [base object] and augment by all possible [augmentations]

11 Search by Augmentations While generating [combinatorial object] we start at [base object] and augment by all possible [augmentations] but keep in mind [symmetries].

12 Search by Augmentations While generating [combinatorial object] we start at [base object] and augment by all possible [augmentations] but keep in mind [symmetries]. This technique leads to an isomorphism class appearing once for every possible augmentation sequence that generates that unlabeled object.

13 Search by Augmentations While generating [combinatorial object] we start at [base object] and augment by all possible [augmentations] but keep in mind [symmetries]. This technique leads to an isomorphism class appearing once for every possible augmentation sequence that generates that unlabeled object. So, we define a canonical deletion which is an invariant reversal of the augmentation.

14 Search by Augmentations While generating [combinatorial object] we start at [base object] and augment by all possible [augmentations] but keep in mind [symmetries]. This technique leads to an isomorphism class appearing once for every possible augmentation sequence that generates that unlabeled object. So, we define a canonical deletion which is an invariant reversal of the augmentation. Stick in results from [your favorite combinatorics], and you may have an efficient algorithm!

15 Search by Augmentations While generating triangle free graphs we start at an isolated vertex and augment by all possible vertices (with independent neighbors) but keep in mind set orbits. This technique leads to an isomorphism class appearing once for every possible augmentation sequence that generates that unlabeled object. So, we define a canonical deletion which is an invariant reversal of the augmentation. Stick in results from graph theory, and you may have an efficient algorithm!

16 Generating with Vertex Augmentations

17 Generating with Vertex Augmentations

18 Generating with Vertex Augmentations

19 Generating with Vertex Augmentations

20 Generating with Vertex Augmentations

21 Generating with Vertex Augmentations

22 Search as a Poset Say H G if G is reachable from H via a sequence of augmentations. This defines a partial order on partial objects.

23 Search as a Poset

24 Search as a Poset

25 Search as a Poset

26 Search as a Poset

27 Search as a Poset

28 Search as a Poset

29 Search as a Poset

30 Canonical Labeling A canonical labeling takes a labeled graph G

31 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G)

32 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G) so that any H = G with labels σ H (v)

33 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G) so that any H = G with labels σ H (v) has an isomorphism σ 1 H (σ G(v)) from G to H.

34 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G) so that any H = G with labels σ H (v) has an isomorphism σ 1 H (σ G(v)) from G to H.

35 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G) so that any H = G with labels σ H (v) has an isomorphism σ 1 H (σ G(v)) from G to H.

36 Canonical Labeling A canonical labeling takes a labeled graph G and applies labels σ G (v) to each v V (G) so that any H = G with labels σ H (v) has an isomorphism σ 1 H (σ G(v)) from G to H. Canonical labels can be computed by McKay s nauty software.

37 Brendan McKay Isomorph-free exhaustive generation 258 citations on Google Scholar.

38 How it works Remove isomorphs by:

39 How it works Remove isomorphs by: 1. Define a canonical deletion (the augmentation that should have generated this graph).

40 How it works Remove isomorphs by: 1. Define a canonical deletion (the augmentation that should have generated this graph). 2. The canonical deletion must be invariant.

41 How it works Remove isomorphs by: 1. Define a canonical deletion (the augmentation that should have generated this graph). 2. The canonical deletion must be invariant. 3. Reject any augmentations not isomorphic to canonical deletion.

42 How it works Remove isomorphs by: 1. Define a canonical deletion (the augmentation that should have generated this graph). 2. The canonical deletion must be invariant. 3. Reject any augmentations not isomorphic to canonical deletion. 4. Optimization: Use deletion rule to reduce number of attempted augmentations (e.g. minimum degree).

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46 (take a drink)

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49 Isomorph-free Generation Examples 1. Triangle-Free Graphs

50 Isomorph-free Generation Examples 1. Triangle-Free Graphs 2. Posets (up to order 16)

51 Isomorph-free Generation Examples 1. Triangle-Free Graphs 2. Posets (up to order 16) 3. Latin Squares

52 Isomorph-free Generation Examples 1. Triangle-Free Graphs 2. Posets (up to order 16) 3. Latin Squares 4. Steiner Triple Systems

53 Isomorph-free Generation Examples 1. Triangle-Free Graphs 2. Posets (up to order 16) 3. Latin Squares 4. Steiner Triple Systems 5. Verify Reconstruction Conjecture (up to 11 vertices).

54 Examples in Software Canonical augmentation appears in the following software: 1. McKay s geng and genbg programs. 2. Sage.org s Graph library: graphs() and digraphs() methods.

55 Augmentations by Edges

56 Augmentations by Edges

57 Augmentations by Edges

58 Augmentations by Edges

59 Augmentations by Edges

60 Augmentations by Edges

61 Examples in Software Canonical augmentation by ears appears in the following software: 1. Sage.org s Graph library: graphs() and digraphs() methods. (Use the flag augment= edges )

62 Augmentations by Ears All 2-connected graphs have an ear decomposition.

63 Augmentations by Ears All 2-connected graphs have an ear decomposition.

64 Augmentations by Ears All 2-connected graphs have an ear decomposition.

65 Augmentations by Ears All 2-connected graphs have an ear decomposition.

66 Augmentations by Ears All 2-connected graphs have an ear decomposition.

67 Ear Augmentation Applications

68 Ear Augmentation Applications 1. Edge-Reconstruction Conjecture.

69 Ear Augmentation Applications 1. Edge-Reconstruction Conjecture. 2. Extremal graphs with a fixed number of perfect matchings.

70 Ear Augmentation Applications 1. Edge-Reconstruction Conjecture. 2. Extremal graphs with a fixed number of perfect matchings. 3. Uniquely K r -saturated graphs.

71 Uniquely K r -Saturated Graphs Definition A graph G is uniquely K r -saturated if G contains no K r and for every edge e G admits exactly one copy of K r in G + e.

72 Uniquely K r -Saturated Graphs Definition A graph G is uniquely K r -saturated if G contains no K r and for every edge e G admits exactly one copy of K r in G + e. (a) 1-book (b) 2-book (c) 3-book Figure: The (r 2)-books are uniquely K r saturated.

73 Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r+1 -saturated graph. Removing a dominating vertex from a uniquely K r -saturated graph creates a uniquely K r 1 -saturated graph.

74 Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r+1 -saturated graph. Removing a dominating vertex from a uniquely K r -saturated graph creates a uniquely K r 1 -saturated graph. Q: Which uniquely K r -saturated graphs have no dominating vertex?

75 Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r+1 -saturated graph. Removing a dominating vertex from a uniquely K r -saturated graph creates a uniquely K r 1 -saturated graph. Q: Which uniquely K r -saturated graphs have no dominating vertex? A: Known for r {2, 3}.

76 Joshua Cooper Paul Wenger Two Conjectures: 1. For each r, there are a finite number of uniquely K r -saturated graphs with no dominating vertex. 2. For each r, every uniquely K r -saturated graph with no dominating vertex is regular. Previously verified to 9 vertices.

77 Uniquely K r -Saturated Graphs 1. Uniquely K r -saturated graphs have diameter 2 (and are 2-connected). 2. Strength: K 4 -free is a sparse, monotone property. 3. Verified for r = 4 and n Verified for r {5, 6} and n 11.

78 (a) (b) (c) The only uniquely K 4 -saturated graphs on up to 12 vertices with no dominating vertex.

79 Coupled Augmentations Idea: Let the problem constraints dictate the augmentation type.

80 Caveat Programmer Balance: Number of Nodes vs. Computation per Node

81 Caveat Programmer Balance: Number of Nodes vs. Computation per Node

82 Caveat Programmer Balance: Number of Nodes vs. Computation per Node

83 K r -completions Consider searching for uniquely K r -saturated graphs.

84 K r -completions Consider searching for uniquely K r -saturated graphs. Every non-edge requires a K r completion.

85 K r -completions Consider searching for uniquely K r -saturated graphs. Every non-edge requires a K r completion. Augmentation: Pick a vertex pair to be completed non-edge, also select where to place the Kr completion.

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92 Canonical Non-Edge A deletion requires picking a canonical completed non-edge.

93 Canonical Non-Edge A deletion requires picking a canonical completed non-edge. Remove all edges which came from that edge s completion.

94 Canonical Non-Edge A deletion requires picking a canonical completed non-edge. Remove all edges which came from that edge s completion. But only if they don t appear in another completion!

95 Canonical Non-Edge A deletion requires picking a canonical completed non-edge. Remove all edges which came from that edge s completion. But only if they don t appear in another completion! Extra Step: Try filling all open pairs with edges.

96 Results (d) (e) Joint with Stephen G. Hartke (N = 13) The only uniquely K 4 -saturated graph on 14 vertices is the 2-book with 12 pages.

97 To learn more... B. D. McKay. Isomorph-free exhaustive generation. B. D. McKay. Small graphs are reconstructible. D. Stolee. Isomorph-free generation of 2-connected graphs with applications. D. Stolee. Generating p-extremal graphs. F. Margot. Pruning by isomorphism in branch-and-cut. B. D. McKay, A. Meynert. Small latin squares, quasigroups, and loops. G. Brinkmann, B. D. McKay. Posets on up to 16 points. P. Kaski, P. R. J. Östergard. The Steiner triple systems of order 19.

98 The canonical augmentation method Derrick Stolee University of Nebraska Lincoln s-dstolee1/ May 13, Supported by NSF grants CCF and DMS