Computational Methods for Domination Problems by William Herbert Bird B.Sc., University of Victoria, 2011 M.Sc., University of Victoria, 2013 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Computer Science c William Herbert Bird, 2017 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Computational Methods for Domination Problems by William Herbert Bird B.Sc., University of Victoria, 2011 M.Sc., University of Victoria, 2013 Supervisory Committee Dr. Wendy Myrvold, Supervisor (Department of Computer Science) Dr. Venkatesh Srinivasan, Departmental Member (Department of Computer Science) Dr. Kieka Mynhardt, Outside Member (Department of Mathematics and Statistics) iii Supervisory Committee Dr. Wendy Myrvold, Supervisor (Department of Computer Science) Dr. Venkatesh Srinivasan, Departmental Member (Department of Computer Science) Dr. Kieka Mynhardt, Outside Member (Department of Mathematics and Statistics) ABSTRACT For a graph G, the minimum dominating set problem is to find a minimum size set S of vertices of G such that every vertex is either in S or adjacent to a vertex in the set. The decision version of this problem, which asks whether G has a dominating set of a particular size k, is known to be NP-complete, and no polynomial time algorithm to solve the problem is currently known to exist. The queen domination problem is to find the minimum number of queens which, collectively, can attack every square on an n × n chess board. The related border queen problem is to find such a collection of queens with the added restriction that all queens lie on the outer border of the board. This thesis studies practical exponential time algorithms for solving domination problems, and presents an experimental comparison of several different algorithms, with the goal of producing a broadly effective iv general domination solver for use by future researchers. The developed algorithms are then used to solve several open problems, including cases of the queen domination problem and the border queen problem. In addition, new theoretical upper bounds are presented for the border queen problem for some families of queen graphs. v Contents Supervisory Committee ii Abstract iii Table of Contents v List of Tables viii List of Figures xi List of Algorithms xiv 1 Introduction 1 1.1 Definitions . 2 1.1.1 Graphs . 3 1.1.2 Dominating Sets . 3 1.1.3 Independent Dominating Sets . 4 1.2 Complexity and Parameterized Complexity . 5 1.3 Related Computational Problems . 9 1.4 Algorithms to Compute Minimum Dominating Sets . 11 2 Queen Graphs and Other Interesting Graph Classes 15 2.1 The Queen Domination Problem . 16 2.2 Irredundant Sets . 18 vi 2.3 The Border Queen Problem . 20 2.4 Kneser Graphs . 26 2.5 Covering Codes and Football Pools . 26 2.6 Triangle Grid Graphs . 30 2.7 Hex Rook Graphs . 31 2.8 Cartesian Products of Cycles . 32 3 Algorithms 34 3.1 Backtracking Framework . 35 3.1.1 Bounding Condition . 37 3.1.2 Vertex Selection . 37 3.1.3 Neighbour Ordering . 37 3.2 Bounding With Fixed Vertex Ordering . 38 3.2.1 Implementation: Algorithm 3.2 . 38 3.3 Domination Degree Algorithms . 41 3.3.1 Domination Degree Multiset . 44 3.3.2 Candidate Degree Priority Queue . 47 3.3.3 Implementation: Algorithm 3.5 . 58 3.4 Max Dominator Degree Algorithms . 61 3.4.1 MDD Ranking Data Structure . 63 3.4.2 Implementation: Algorithm 3.7 . 66 4 Experimental Evaluation of Domination Algorithms 70 4.1 Input Graph Dataset . 72 4.2 Methodology . 73 4.2.1 Mitigating the impact of `luck' . 75 4.3 Fixed-Ordering Implementations . 77 vii 4.4 Domination Degree Implementations . 81 4.4.1 Single Aspect Comparisons . 83 4.5 Max Dominator Degree Implementations . 88 4.5.1 Single Aspect Comparison . 88 4.6 Comparison of Framework Algorithms . 96 4.7 Comparison with SageMath . 100 4.8 Choosing Representative Algorithms . 103 4.8.1 Overall Variant Comparison . 105 4.8.2 Comparison of Variants by Graph Family . 107 5 New Domination Results for Queen Problems 113 5.1 Computing Independent Dominating Sets . 114 5.2 Splitting Computation Among Processes . 115 5.3 Counting Solutions up to Isomorphism . 118 5.4 Certificates of Independent Dominating Sets . 119 5.5 Rotated Border Constructions . 122 5.5.1 Searching for Minimum RBCs . 139 5.5.2 Summary of Border Queen Results . 142 6 Unidom 149 6.1 The unidom Architecture . 149 6.2 Input Source . 151 6.3 Preprocessing Filters . 153 6.4 Solver . 156 6.5 Output Proxy . 158 7 Conclusions and Future Research 161 viii Bibliography 165 ix List of Tables Table 2.1 Domination Numbers of Kneser Graphs . 27 Table 2.2 Solutions to the Football Pool Problem for n = 1;:::; 10 . 29 Table 2.3 Domination Numbers of Hypercubes Qn for n = 1;:::; 11 . 29 Table 2.4 Domination Numbers of Hex Rook Graphs . 32 Table 4.1 Optimization Experiment Input Graphs . 73 Table 4.2 Fixed Ordering Running Times: Covering Code Graphs . 79 Table 4.3 Fixed Ordering Running Times: Hex Rook Graphs . 79 Table 4.4 Fixed Ordering Running Times: Kneser Graphs . 79 Table 4.5 Fixed Ordering Running Times: Knight Graphs . 80 Table 4.6 Fixed Ordering Running Times: Cartesian Products of Cycles . 80 Table 4.7 Fixed Ordering Running Times: Queen Graphs . 80 Table 4.8 Fixed Ordering Running Times: Triangle Grid Graphs . 81 Table 4.9 DD Bounding: Summary of maximum time ratios for all aspects on all graph families . 85 Table 4.10 DD Bounding: Summary of maximum total call ratios for all aspects on all graph families . 88 Table 4.11 MDD Bounding: Summary of maximum time ratios for all aspects on all graph families . 95 Table 4.12 MDD Bounding: Summary of maximum total call ratios for all aspects on all graph families . 96 x Table 4.13 Comparison of Framework Algorithms - Maximum Times: Covering Code Graphs . 97 Table 4.14 Comparison of Framework Algorithms - Maximum Times: Hex Rook Graphs . 98 Table 4.15 Comparison of Framework Algorithms - Maximum Times: Kneser Graphs 98 Table 4.16 Comparison of Framework Algorithms - Maximum Times: Knight Graphs 99 Table 4.17 Comparison of Framework Algorithms - Maximum Times: Cartesian Products of Cycles . 99 Table 4.18 Comparison of Framework Algorithms - Maximum Times: Queen Graphs 99 Table 4.19 Comparison of Framework Algorithms - Maximum Times: Triangular Grid Graphs . 100 Table 4.20 SageMath vs. Framework 3.1 - Maximum Times: Covering Code Graphs101 Table 4.21 SageMath vs. Framework 3.1 - Maximum Times: Hex Rook Graphs . 101 Table 4.22 SageMath vs. Framework 3.1 - Maximum Times: Kneser Graphs . 101 Table 4.23 SageMath vs. Framework 3.1 - Maximum Times: Knight Graphs . 102 Table 4.24 SageMath vs. Framework 3.1 - Maximum Times: Cartesian Products of Cycles . 102 Table 4.25 SageMath vs. Framework 3.1 - Maximum Times: Queen Graphs . 102 Table 4.26 SageMath vs. Framework 3.1 - Maximum Times: Triangular Grid Graphs102 Table 4.27 Best 10 average maximum time fractions of tested algorithms on the entire input dataset. 106 Table 4.28 Best 10 average maximum time fractions of tested algorithms on the 20 moderately difficult graphs in the input dataset. 106 Table 4.29 Maximum times of Framework 3.1 variants on Covering Code graphs. 108 Table 4.30 Maximum times of Framework 3.1 variants on Hex Rook graphs. 109 Table 4.31 Maximum times of Framework 3.1 variants on Kneser graphs. 110 xi Table 4.32 Maximum times of Framework 3.1 variants on Knight graphs. 110 Table 4.33 Maximum times of Framework 3.1 variants on Cartesian Products of Cycles. 111 Table 4.34 Maximum times of Framework 3.1 variants on Queen graphs. 111 Table 4.35 Maximum times of Framework 3.1 variants on Triangular Grid graphs. 112 Table 5.1 Domination Numbers of Queen Graphs . 114 Table 5.2 Number of Minimum Dominating Sets of Queen Graphs up to Isomor- phism . 118 Table 5.3 Number of Minimum Independent Dominating Sets of Queen Graphs up to Isomorphism . 119 Table 5.4 Minimum Border Dominating Sets of Queen Graphs up to Isomorphism 119 Table 5.5 Summary of Border Domination Parameters . 143 xii List of Figures Figure 2.1 3 × 3 chess board and Queen graph of order 3 . 17 Figure 2.2 A maximal irredundant set of C5 .................... 19 Figure 2.3 Examples of the construction used in the proof of Theorem 2.8. 23 Figure 2.4 Counterexample of an assertion by Burchett in [9]. 24 Figure 2.5 Triangle grid and hex rook graphs of order 3 . 30 Figure 3.1 An 8 × 8 board with two queens. 42 Figure 3.2 A 10 × 10 board with three queens. 61 Figure 4.1 DD Bounding: Histogram of pairwise ratios for Min. CD vs. Max. CD vertex selection. 86 Figure 4.2 DD Bounding: Histogram of pairwise ratios for ascending vs. descend- ing neighbour order. 86 Figure 4.3 DD Bounding: Histogram of pairwise ratios for force stop optimization disabled vs. enabled. 87 Figure 4.4 DD Bounding: Histogram of pairwise ratios for bound rechecking op- timization disabled vs. enabled. 87 Figure 4.5 MDD Bounding: Histogram of pairwise ratios for Min. CD vs. Max. CD vertex selection. 90.