Public Defence. Jan Goedgebeur
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1 Generation Algorithms for Mathematical and Chemical Problems Public Defence Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics, Computer Science and Statistics Ghent University caagt
2 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes
3 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes
4 What is a graph? Set of vertices
5 What is a graph? Set of vertices Set of edges
6 Why are graphs useful? Modelling road networks
7 Why are graphs useful? Modelling road networks
8 Why are graphs useful? Modelling road networks
9 Why are graphs useful? Modelling road networks
10 Why are graphs useful? Modelling molecules Benzene (C 6 H 6 ) H H C C C H H C C C H H
11 Why are graphs useful? Modelling molecules Benzene (C 6 H 6 )
12 Why are graphs useful? Determine assignment of teachers to courses
13 Why are graphs useful? Determine assignment of teachers to courses
14 Why are graphs useful? Determine assignment of teachers to courses
15 Why are graphs useful? Determine assignment of teachers to courses
16 Degree of a vertex Degree = number of neighbours
17 Degree of a vertex Degree = number of neighbours k-regular = every vertex has degree k 3-regular = cubic 3-regular graph (Petersen Graph)
18 Isomorphism What are isomorphic graphs? graphs which have the same structure
19 Isomorphism What are isomorphic graphs? graphs which have the same structure
20 Isomorphism What are isomorphic graphs? graphs which have the same structure
21 Isomorphism What are isomorphic graphs? graphs which have the same structure
22 Planarity A graph is planar if it can be drawn in the plane without crossing edges.
23 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes
24 What is an algorithm? An algorithm is a step-by-step procedure to solve a problem.
25 What is an algorithm? An algorithm is a step-by-step procedure to solve a problem. Example: recipe to bake a cake
26 What is an algorithm? An algorithm is a step-by-step procedure to solve a problem. Example: recipe to bake a cake Efficiency of algorithms...
27 What is structure generation? Structure generation: generate all structures from a given class. Note: all structures without isomorphic copies. Examples: Generate a list of all graphs with 10 vertices. Generate a list of all cubic graphs with 20 vertices. Generate a list of all molecules for the formula C 20 H 10.
28 What is a generator? Example: generator for odd numbers
29 What is a generator? Example: generator for odd numbers
30 What is a generator? Example: generator for odd numbers
31 What is a generator? Example: generator for odd numbers
32 What is a generator? Example: generator for odd numbers
33 What is a generator? Example: generator for odd numbers
34 What is a generator? Example: generator for odd numbers Not isomorphism-free (duplicates) Not exhaustive (9 missing) Our generators are exhaustive and isomorphism-free
35 Why structure generation? Why is structure generation useful? Applications in amongst others: Mathematics Chemistry
36 Applications in Mathematics Used to test mathematical conjectures. E.g.: all cubic graphs have property X Generate cubic graphs and test if they have this property.
37 Applications in Chemistry Used for structure elucidation. Goal: identifying molecules in a substance.
38 Structure elucidation Mass spectrum of a substance
39 Structure elucidation
40 Applications in other areas Used to determine the best pitch sequence for a tire. The surface of a tire can consist of several types of blocks, e.g. short, medium and long ones. The sequence of these blocks is called the pitch sequence.
41 What is graph generation? Construct all non-isomorphic graphs with n vertices of a given graph class. For example: all graphs cubic graphs planar graphs...
42 Naive generator for cubic graphs Great idea for a generator for cubic graphs: Use existing generator to generate all graphs. Test the generated graphs and only output the cubic ones.
43 Naive generator for cubic graphs Great idea for a generator for cubic graphs: Use existing generator to generate all graphs. Test the generated graphs and only output the cubic ones. Problem: Only % of the graphs with 12 vertices are cubic... So for interesting graph classes it is justified to develop specialised algorithms.
44 Naive generator for cubic graphs Great idea for a generator for cubic graphs: Use existing generator to generate all graphs. Test the generated graphs and only output the cubic ones. Problem: Only % of the graphs with 12 vertices are cubic... So for interesting graph classes it is justified to develop specialised algorithms.
45 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes Joint work with Prof. Brendan McKay (Australian National University)
46 Cubic graphs Cubic or 3-regular graphs are graphs where every vertex has degree 3 (3 edges meet at each vertex).
47 Cubic graphs They are very interesting in chemistry as models of molecules (vertices are e.g. carbon atoms as in fullerenes). They are very interesting in mathematics (for a lot of conjectures the smallest possible counterexamples are cubic graphs).
48 History of constructing cubic graphs The first construction method for cubic graphs dates from The next ones date from: 1966, 1968, 1971, 1974, 1976, 1985, 1992, 1999, People who worked on constructing cubic graphs: De Vries, Balaban, Bussemaker, Seidel, Imrich, Petrenjuk, Cobeljic, Cvetkovic, Faradžev, McKay, Royle, Brinkmann, Meringer, Sanjmyatav.
49 Generating cubic graphs Our generator: The following operation can be used to construct cubic graphs: We call this the edge operation.
50 Example: generating cubic graphs
51 Example: generating cubic graphs
52 Example: generating cubic graphs
53 Example: generating cubic graphs
54 Example: generating cubic graphs
55 Example: generating cubic graphs
56 Example: generating cubic graphs
57 Example: generating cubic graphs
58 Example: generating cubic graphs
59 Example: generating cubic graphs
60 Example: generating cubic graphs
61 How can isomorphic copies occur? A parent graph can have isomorphic children.
62 How can isomorphic copies occur? A graph can be constructed from different parents.
63 How to avoid generating isomorphic copies? Isomorphism rejection by lists
64 How to avoid generating isomorphic copies? Isomorphism rejection by lists Keep list of non-isomorphic graphs which were generated so far. Only accept graphs which were not generated before.
65 How to avoid generating isomorphic copies? Isomorphism rejection by lists Keep list of non-isomorphic graphs which were generated so far. Only accept graphs which were not generated before. Not feasible for large lists...
66 How to avoid generating isomorphic copies? Isomorphism rejection by lists Keep list of non-isomorphic graphs which were generated so far. Only accept graphs which were not generated before. Not feasible for large lists... McKay s canonical construction path method Read/Faradžev-type orderly algorithms Double coset method Homomorphism principle...
67 Results cubic graphs n # graphs time graphs/s 4 1 0s 6 2 0s 8 5 0s s s s s s s s s h h days days More than 4 times faster than previously fastest generator for cubic graphs.
68 Results Personal computer Jan
69 , Results Supercomputer Ghent University [Photo: Kenneth Hoste]
70 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes
71 What are snarks? Lewis Caroll The Hunting of the Snark (1876)
72 What are snarks? Mathematical definition introduced by Gardner in Definition: Always: cubic graph no triangles strongly connected not 3-edge-colourable Sometimes: no squares
73 Edge-colourability Proper edge colouring: Colouring of the edges of the graph such that the edges of every vertex all have a different colour.
74 Edge-colourability Proper edge colouring: Colouring of the edges of the graph such that the edges of every vertex all have a different colour. Invalid edge colouring Proper edge colouring
75 Edge-colourability Proper edge colouring: Colouring of the edges of the graph such that the edges of every vertex all have a different colour. Invalid edge colouring Proper edge colouring For cubic graphs 3 or 4 colours are required for a proper edge colouring. Snarks do not have a proper edge colouring with 3 colours.
76 What are snarks? Snarks are very rare. For 28 vertices only % of the cubic graphs without triangles are snarks. and the rate is decreasing as the number of vertices increases... Smallest snark: Petersen graph.
77 Why are snarks useful? Snarks are often candidates for smallest counterexamples... For a lot of important conjectures it is proven that: If the conjecture is false there is a snark that is a counterexample. very often: the smallest counterexample is a snark.
78 Why are snarks useful? Snarks are often candidates for smallest counterexamples... For a lot of important conjectures it is proven that: If the conjecture is false there is a snark that is a counterexample. very often: the smallest counterexample is a snark. E.g.: Cycle double cover conjecture
79 Cycle double cover conjecture Conjecture (cycle double cover, 1973) Every bridgeless graph has a collection of cycles such that each edge of the graph is contained in exactly two of the cycles.
80 Cycle double cover conjecture Conjecture (cycle double cover, 1973) Every bridgeless graph has a collection of cycles such that each edge of the graph is contained in exactly two of the cycles.
81 Why are snarks useful? Smallest possible counterexamples to the cycle double cover conjecture are snarks...
82 Why are snarks useful? Smallest possible counterexamples to the cycle double cover conjecture are snarks... So if you want to know whether this conjecture is true or not for the more than graphs up to 18 vertices, just test the 3 snarks.
83 Generation of snarks Naive filter approach: Generate all cubic graphs and test if they are 3-edge-colourable. That is what existing generators do. Our approach: Prune parent graphs that cannot lead to snarks.
84 Results snarks Comparison with fastest generator for snarks so far: Snarks: 24 vertices: speedup: snarks 26 vertices: speedup: snarks 28 vertices: speedup: snarks Snarks without squares: 24 vertices: speedup: snarks 26 vertices: speedup: snarks 28 vertices: speedup: snarks
85 Number of snarks n snarks snarks without squares ?
86 Counterexamples to conjectures Snarks are a source for possible counterexamples to conjectures... Can be used to...? = Could in principle be used to...? = Is it really useful for...?
87 Counterexamples to conjectures Collaboration with Prof. Klas Markström and Dr. Jonas Hägglund (Umeå University, Sweden). We used the snarks to test 22 published conjectures.
88 Counterexamples to conjectures Collaboration with Prof. Klas Markström and Dr. Jonas Hägglund (Umeå University, Sweden). We used the snarks to test 22 published conjectures. Found counterexamples to 8 conjectures of among others: Prof. Bill Jackson (Queen Mary University of London) Prof. Cun-Quan Zhang (West Virginia University)
89 Counterexamples to conjectures A minimal counterexample to 4 conjectures: Snark with 34 vertices.
90 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes
91 , Ramsey numbers Joint work with Prof. Stanislaw Radziszowski (Rochester Institute of Technology, NY, USA)
92 What are Ramsey numbers? The Ramsey number R(k, l) is the solution to the party problem: What is the minimum number of guests that must be invited to a party so that it is guaranteed that: or at least k people will be mutual acquaintances at least l people will be mutual strangers
93 Ramsey numbers example Example: R(3, 3) =? Red edge: are acquaintances Blue edge: are strangers
94 Ramsey numbers example Example: R(3, 3) =? Red edge: are acquaintances Blue edge: are strangers R(3, 3) 6
95 Results Ramsey numbers Ramsey numbers are very difficult to determine... k R(3, k) k R(3, k) (old: 43) (old: 51) (old: 69) (old: 78) (old: 88) (old: 99) Values and bounds for Ramsey numbers R(3, k).
96 Results Ramsey numbers Ramsey numbers are very difficult to determine... k R(3, k) k R(3, k) (old: 43) (old: 51) (old: 69) (old: 78) (old: 88) (old: 99) Values and bounds for Ramsey numbers R(3, k). This was the first improvement of the upper bounds of triangle Ramsey numbers since 25 years.
97 Outline Generation Algorithms for Mathematical and Chemical Problems 1 Introduction Graph theory Generation algorithms 2 Mathematical and Chemical problems Cubic graphs Snarks Ramsey numbers Fullerenes Joint work with Prof. Brendan McKay (Australian National University)
98 , Warning May contain traces of kangaroo...
99 C 60 buckyball Discovered by Prof. Kroto et al. in Nobel Prize in Chemistry. Sir Harold Kroto. Article in Nature citations (and counting).
100 Fullerenes Fullerenes (mathematically) A fullerene is a cubic planar graph with faces of size 5 and 6. Euler s formula implies that there are exactly 12 pentagonal faces.
101 C 60 buckyball
102 C 60 buckyball
103 C 60 buckyball - 2D
104 C 60 buckyball
105 C 660
106 Carbon nanotubes
107 Carbon nanotubes [Source: Het Laatste Nieuws, 4 February 2013]
108 History of generation of fullerenes 1991: Manolopoulos et al. 1991: Liu et al. 1993: Sah 1995: Yoshida and Osawa But all incomplete or inefficient...
109 History of generation of fullerenes 1991: Manolopoulos et al. 1991: Liu et al. 1993: Sah 1995: Yoshida and Osawa But all incomplete or inefficient : Brinkmann and Dress Complete Efficient (program: fullgen)
110 Construction operations
111 Construction operations An L 0 expansion:
112 Results fullerenes Program based on this algorithm: More than 3.5 times faster than previously fastest generator (i.e. fullgen).
113 Results fullerenes Program based on this algorithm: More than 3.5 times faster than previously fastest generator (i.e. fullgen). Contradicting results with fullgen lead to the detection of a (now fixed) bug in fullgen. Missed fullerenes starting from 136 vertices.
114 More results Used our program to generate all non-isomorphic fullerenes up to 400 vertices. Independently confirmed by fullgen (new version) up to 380 vertices. This allowed us to determine the smallest fullerene that does not have a face spiral (an open problem since 1991)...
115 Face spirals What is a face spiral?
116 Face spirals What is a face spiral? Positions of the pentagons: 1, 7, 9, 11, 13, 15, 18, 20, 22, 24, 26, 32.
117 Face spirals The International Union of Pure and Applied Chemistry recommended face spirals as a basis for fullerene nomenclature... but unfortunately not every fullerene has a face spiral...
118 Face spirals A face spiral which does not work:
119 Face spirals The smallest fullerene without a face spiral: Has 380 vertices.
120 Thanks for your attention!
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