Detecting Non-Hamiltonian Graphs by Improved Linear Programs and Graph Reductions A thesis submitted for the degree of Doctor of Philosophy Kieran Clancy B.Sc. (Hons) School of Computer Science, Engineering & Mathematics Faculty of Science & Engineering Flinders University May 2017 To my mother Gina for her immeasurable support. Contents Contentsi List of Tables iv List of Figures vii Summaryx Declaration xiii Acknowledgements xiv Glossary of terms xviii Archive of problem sets and algorithms xx 1 Introduction and background1 1.1 Hamiltonian cycle problem....................1 1.2 Graph theory...........................6 1.2.1 Hamiltonian cycle problem for cubic graphs...... 11 1.3 Linear programming relaxations................. 12 2 Identifying non-Hamiltonian graphs by linear programming 15 2.1 Existing models to solve TSP and HCP............. 17 i Contents ii 2.1.1 Subtour elimination model and MCF.......... 17 2.1.2 Tightened multi-commodity flow model......... 21 2.1.3 SST model......................... 23 2.1.4 The Base Model...................... 25 2.2 Comparisons of LP models.................... 28 2.2.1 Adapting TSP models to solve HCP.......... 29 2.2.2 Results of LP models on HCP instances........ 32 2.2.3 Adapting the Base Model to solve TSP......... 34 2.2.4 Generating TSP instances based on cubic graphs... 37 2.2.5 Results of LP models on TSP instances......... 42 2.2.6 A conjecture on the strength of the Base Model.... 44 2.3 Classifications of difficult cubic graphs............. 51 2.3.1 Vertex and edge connectivity.............. 51 2.3.2 Graph toughness..................... 55 2.4 Concluding remarks on the Base Model............. 62 3 Hamiltonicity-preserving graph reductions 64 3.1 Graph reductions based on subgraphs.............. 70 3.2 Graph reductions based on Hamiltonian and non-Hamiltonian edges................................ 74 3.3 Edge orbits and their classification............... 79 3.4 Graph reductions based on edge orbits............. 92 3.5 Graph reduction algorithm.................... 98 3.6 Results of reduction algorithm on cubic graphs......... 110 4 Extending the Base Model 120 4.1 Merging SST with the Base Model............... 125 Contents iii 4.1.1 Base-SST model with multiple starting vertices.... 127 4.1.2 Extended Base-SST model................ 130 4.2 Constraints involving forced edges................ 142 4.3 Constraints based on 3-cuts................... 149 4.4 Constraints based on an eigenvalue of Hamiltonian permuta- tion matrices........................... 152 4.5 Results of combined extensions................. 162 4.6 Detecting non-Hamiltonicity of graphs by using LP models on their subgraphs.......................... 173 5 Conclusions and future work 176 5.1 Summary of results........................ 177 5.2 Future work arising from Chapter 2............... 178 5.3 Future work arising from Chapter 3............... 179 5.4 Future work arising from Chapter 4............... 179 Appendix A Non-Hamiltonian non-bridge cubic graph sets 182 A.1 NHNB20 GENREG IDs..................... 183 A.2 NHNB20PR edge lists...................... 185 Appendix B ATSP problem sets 189 Appendix C Implementation of graph reduction algorithm 204 Bibliography 220 List of Tables 2.5 Results of MCF, MCF+, SST and the Base Model on non- Hamiltonian graphs up to order 20............... 33 2.6 Results of MCF, MCF+, SST and the Base Model on NHNB20 34 2.9 Results of MCF, MCF+, SST and the Base Model on ATSP16A 43 2.10 Results of MCF, MCF+, SST and the Base Model on ATSP16AC 43 2.13 Hamiltonicity of connected 10-vertex graphs up to isomor- phism by vertex connectivity................... 53 2.14 The numbers of non-Hamiltonian and Hamiltonian cubic graphs up to order 20........................... 54 2.15 The percentage of cubic graphs up to order 20 that are Hamil- tonian............................... 54 2.16 Contingency table for cubic graphs up to order 20 by Hamil- tonicity and connectivity..................... 55 2.17 Contingency table for non-Hamiltonian cubic graphs up to order 20 by Base Model feasibility and connectivity...... 55 2.19 Hamiltonian and non-Hamiltonian cubic graphs up to order 20 by toughness.......................... 59 2.20 Base Model feasibility for non-Hamiltonian cubic graphs up to order 20 by toughness...................... 60 iv List of Tables v 3.7 Number of asymmetric and non-asymmetric cubic graphs by order up to 20 vertices...................... 81 3.12 Percentage of edges in each type of orbit, as classified by The- orem 3.36, for non-asymmetric cubic graphs up to order 20.. 90 3.18 Automorphism group size of cubic graphs of order between 6 and 20 with at least one triangle................. 100 3.19 Cubic graphs up to order 18 where multiple graph reductions of the types star, pinwheel, cycle and cut were simultaneously applicable............................. 102 3.20 The number of comparisons made between each pair of reduc- tions................................ 102 3.21 The number of comparisons between reductions where one re- duction led to a graph with fewer edges than the other reduction103 3.24 Number of cubic graphs up to order 20 that are reducible by Algorithm 3.1........................... 110 3.25 Number of Hamiltonian cubic graphs up to order 20 that are reducible by Algorithm 3.1.................... 111 3.26 Number of graphs in NHNB20 that are reducible by Algo- rithm 3.1.............................. 111 3.27 Base Model feasibility of reducible instances of NHNB20, be- fore and after reduction...................... 112 3.28 Base Model feasibility after reduction versus Base Model fea- sibility before reduction, for instances of NHNB20 that are partially reduced by Algorithm 3.1............... 113 4.1 Results of the Base Model on NHNB20 and NHNB20PR... 124 4.2 Results of the Base Model on ATSP16A and ATSP16AC... 124 4.3 Results of Base-SST on NHNB20 and NHNB20PR....... 127 List of Tables vi 4.4 Results of Base-SST on ATSP16A and ATSP16AC...... 128 4.5 Results of Base-SST-k on ATSP16A and ATSP16AC..... 130 4.7 Results of Base-SST-k-Ext on ATSP16A and ATSP16AC... 138 4.8 Gaps for each of the Base Model, Base-SST, Base-SST-k and Base-SST-k-Ext on the four instances from ATSP16A for which SST outperforms the Base Model................ 139 4.12 Results of Base-Forced on NHNB20 and NHNB20PR..... 148 4.13 Results of Base-Forced on ATSP16AC............. 148 4.14 Results of Base-3-Cut on NHNB20 and NHNB20PR...... 151 4.15 Results of Base-3-Cut on ATSP16AC.............. 152 4.16 Results of Base-Spectral on NHNB20 and NHNB20PR.... 159 4.17 Results of Base-Spectral on ATSP16A and ATSP16AC.... 160 4.19 Results of Base-Combined on NHNB20 and NHNB20PR... 166 4.21 Results of Base-Combined on ATSP16A and ATSP16AC... 167 4.22 A comparison of Base-Combined with the other extended mod- els on ATSP16A and ATSP16AC................ 168 4.23 Gaps for various models on ATSP16AC instance 31...... 168 4.27 Results of the Base Model and Base-Combined using the sub- graph method on NHNB20 and NHNB20PR.......... 175 5.1 Successive results of the approach described in Section 5.1 on NHNB20.............................. 178 List of Figures 1.1 An example of a Hamiltonian cycle in the 20 vertex dodeca- hedral cubic graph........................2 1.2 The first of Euler's solutions to the knight's tour problem...3 1.3 The Petersen graph and its corresponding adjacency matrix.8 2.1 Visualisation of the feasible region for a relaxed model of HCP 16 2.2 Example of subtours in a TSP instance............. 18 2.3 Example of a subtour elimination constraint.......... 19 2.4 Visualisation of the feasible region after adding additional con- straints............................... 22 2.7 The smallest instance of NHNB20 that induces infeasibility in MCF, MCF+, SST and the Base Model............ 34 2.8 Base Model gaps for randomly generated TSP instances.... 41 2.11 Plot of gaps for the Base Model against gaps for SST..... 45 2.12 The smallest cubic graphs having connectivity 1, 2, and 3... 52 2.18 Selected examples of tough and non-tough cubic graphs.... 58 2.21 The smallest tough cubic graph that induces infeasibility in MCF, MCF+, SST and the Base Model............ 62 3.1 Trivially Hamiltonian and trivially non-Hamiltonian graphs.. 66 3.2 The contraction of a triangle in a graph............. 72 vii List of Figures viii 3.3 A diamond before and after its reduction with diamond .... 74 3.4 A graph with known Hamiltonian edges before and after its reduction with forced ....................... 77 3.5 A graph with a path of degree-2 vertices before and after its reduction with path ....................... 78 3.6 A 6-vertex graph with automorphism group of order 8..... 79 3.8 The Frucht graph, one of the five minimal asymmetric cubic graphs............................... 81 3.9 Vertex and edge orbits for the graph shown in Figure 3.6... 83 3.10 An example of a graph and its line graph............ 85 3.11 Example showing five types of edge orbits in a non-Hamiltonian cubic graph............................ 91 3.13 Examples of incompatible edge sets............... 93 3.14 A graph with known redundant edges before and after its re- duction with star ......................... 95 3.15 A graph with a known Hamiltonian edge and known redundant edges, before and after its reduction with pinwheel ....... 96 3.16 A graph with known redundant edges in a short cycle, before and after its reduction with cycle ................ 97 3.17 A graph with known redundant edges forming a minimal cut set of odd size, before and after its reduction with cut .... 98 3.22 Flowchart of Algorithm 3.1.................... 105 3.23 Flowchart of Algorithms 3.2 to 3.4............... 106 4.6 An example of two directed Hamiltonian cycles showing both possible vertex orderings of four vertices...........