The Quadratic Travelling Salesman Problem: Complexity and Approximation by Brad Woods M.Sc., Simon Fraser University, 2010 B.Math., University of Waterloo, 2007 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science ©Brad Woods 2018 SIMON FRASER UNIVERSITY Summer 2018 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Brad Woods Degree: Doctor of Philosophy (Operations Research) Title: The Quadratic Travelling Salesman Problem: Complexity and Approximation Examining Committee: Chair: Natalia Kouzniak Department of Mathematics Simon Fraser University Abraham Punnen Senior Supervisor Department of Mathematics Simon Fraser University Tamon Stephen Supervisory Committee Member Department of Mathematics Simon Fraser University Binay Bhattacharya Supervisory Committee Member School of Computer Science Simon Fraser University Ramesh Krishnamurti Internal Examiner School of Computer Science Simon Fraser University Donglei Du External Examiner Faculty of Business Administration University of New Brunswick Date Defended: July 12, 2018 ii Abstract In this thesis we study the Quadratic Travelling Salesman Problem (QTSP) which gener- alizes the well-studied Traveling Salesman Problem and several of its variations. QTSP is to find a least cost Hamiltonian cycle in an edge-weighted graph, where costs are defined for all pairs of edges contained in the Hamilton cycle. This is a more general version than the one that appears in the literature as the QTSP, denoted here as the adjacent quadratic TSP, which only considers costs for pairs of adjacent edges. We give a complete character- ization of the QTSP linearization problem and give a polynomial time algorithm to find a linearization whenever one exists. The fixed-rank QTSP is introduced as a restricted version of the QTSP where the cost matrix has fixed rank p. We study QTSP by examining the complexity of searching exponential neighbourhoods for QTSP, the fixed-rank QTSP and the adjacent quadratic TSP. We develop pseudopolynomial time algorithms for many of these special cases, and give FPTAS whenever possible. Polynomial algorithms are given for each special case which is not NP-hard. Keywords: discrete optimization, combinatorial optimization, travelling salesman problem, quadratic travelling salesman problem iii This thesis is dedicated to my parents. iv Acknowledgements I would first and foremost, like to thank my senior supervisor, Dr. Abraham Punnen. With- out his guidance, patience and support throughout my graduate career, this thesis would not have been possible. Dr. Punnen introduced this research topic to me, provided partial results and significant assistance in obtaining some of the results. In addition, I am thankful for the financial support provided through research assistantships. I would like to thank my supervisory committee members, Dr. Tamon Stephen and Dr. Binay Bhattacharya, for their helpful discussions, feedback and comments. Credit is due to Dr. Santosh Kabadi who made significant contributions to the results of Chapter 2. Also, I acknowledge the encouragement and support provided by Dr. Natalia Kouzniak and Dr. Randall Pyke of the Department of Mathematics, SFU Surrey. I owe thanks to everyone at SFU for their help, thoughtful discussions, and friendship. Thank you to all my colleagues in the SFU Optimization Graduate Student Lab, in par- ticular, Xiaorui Li, Pooja Pandey, Piyashat Sripratak, Timothy Yusun, and Yong Zhang. I acknowledge the coding help from Ziyang Zhao. Last, but not least, I would like to thank my friends and family. I could not have done this without your love and support. v Table of Contents Approval ii Abstract iii Dedication iv Acknowledgements v Table of Contents vi List of Tables viii List of Figures ix 1 The quadratic travelling salesman problem 1 1.1 Introduction . 1 1.2 The linear travelling salesman problem . 2 1.3 The quadratic TSP: Basic definitions and notations . 3 1.3.1 Instance representations . 4 1.3.2 Transforming the SQTSP . 5 1.3.3 Transforming the AQTSP . 6 1.3.4 Adjacent-only QTSP . 6 1.3.5 Fixed-rank QTSP . 7 1.4 Motivation . 7 1.5 Literature survey . 8 1.6 Contributions . 8 2 QTSP linearization 10 2.1 Introduction . 10 2.2 The QTSP linearization problem . 11 2.3 Simple sufficient conditions and extensions . 17 3 A class of exponential tours 19 3.1 Introduction . 19 3.2 Single edge ejection tours on G∗ ......................... 20 vi 3.3 Double edge ejection tours on G∗ ........................ 29 3.4 Paired vertex graphs . 34 3.5 Matching edge ejection tours . 38 4 Pyramidal tours 44 4.1 Introduction . 44 4.2 Pyramidal QTSP . 45 4.3 Undirected pyramidal QTSP . 49 5 Halin graphs 52 5.1 Introduction . 52 5.2 Notations and definitions . 52 5.3 Complexity results . 54 5.4 QTSP(A) . 58 5.5 Fixed-rank QTSP . 60 5.6 Multiplicative QTSP . 70 6 k-neighbour TSP on Halin graphs and extensions 72 6.1 Introduction . 72 6.2 Complexity results . 73 6.2.1 TSP(3) .................................. 75 6.2.2 TSP(k) .................................. 87 6.3 TSP(k) on fully reducible graph classes . 88 7 Approximation algorithms and domination analysis 89 7.1 Introduction . 89 7.2 Approximation algorithms . 89 7.2.1 Double-tree heuristic . 90 7.2.2 2-Opt local search . 92 7.3 Domination analysis . 94 7.4 Comparisons against the average . 95 7.4.1 Cheapest insertion . 96 7.4.2 2-Opt local search . 97 7.4.3 Derandomization method . 98 8 Conclusion 100 Bibliography 102 vii List of Tables Table 3.1 Summary of complexity results. 20 Table 6.1 Description of penalty 24-tuple stored at pseudonodes in C. 80 viii List of Figures Figure 1.1 Example graph Gb with n = 6 ..................... 6 Figure 1.2 Optimal TSP tour with respect to length. 8 Figure 1.3 Smoothing of the tour. 8 Figure 1.4 A tour which can be travelled more quickly. 8 ij Figure 2.1 Tours τ ∈ Tn−1 and τ ∈ Tn ...................... 14 Figure 3.1 Example of a G∗ graph . 21 Figure 3.2 An SEE-tour in the graph G∗ given in Figure 3.1 . 21 Figure 3.3 Graph G∗ used in the proof of Theorem 3.2.1 . 22 Figure 3.4 Graph G∗ used in the proof of Theorem 3.2.2 . 23 Figure 3.5 G0 constructed from the graph G∗ given in Figure 3.1. 24 Figure 3.6 A DEE-tour in the graph G∗ given in Figure 3.1 . 30 Figure 3.7 Construction of the graph G∗ used in the proof of Theorem 3.3.2 . 31 Figure 3.8 G0 constructed from the graph G∗ given in Figure 3.1 . 32 Figure 3.9 Example of an optimal expanded cycle D(3) . 34 Figure 3.10 Example of Gp .............................. 34 Figure 3.11 Example of a tour τ in Gp used in the proof of Theorem 3.4.3 . 36 Figure 3.12 Example of a tour τ in Gp used in the proof of Theorem 3.4.4 . 37 Figure 3.13 G0 constructed from the graph Gp given in Figure 3.10 . 37 Figure 3.14 Example graph GM with r = 6 and s = 4 . 38 Figure 3.15 An MEE-tour in the graph GM given in Figure 3.14 . 39 Figure 3.16 Construction used in the proof of Theorem 3.5.1 . 40 Figure 3.17 Construction used in the proof of Theorem 3.5.5 . 42 Figure 3.18 An example of the construction used in the proof of Theorem 3.5.5 42 Figure 4.1 Construction of the graph G∗ used in the proof of Theorem 4.2.2 . 47 Figure 4.2 Construction of the graph G∗ used in the proof of Theorem 4.3.2 . 51 Figure 5.1 A Halin graph with 3 fans . 53 Figure 5.2 A Halin graph H containing fan F . 54 Figure 5.3 The 4-fan gadget . 55 Figure 5.4 Example of the Halin graph constructed from F = C1 ∧ C2 ∧ C3 ∧ C4 56 ix Figure 5.5 Example of a Halin graph G belonging to H(k, 1, 5) . 57 Figure 5.6 Example of a Halin graph in H(4, 2, ∆) used in the proof of Theo- rem 5.3.3 . 59 Figure 5.7 A tour τ in a wheel, which skips edge ci . 60 Figure 5.8 Example of a Halin graph constructed from an instance of PARTI- TION with n = 8............................. 61 Figure 5.9 A tour τ in a wheel, which skips edge ci . 69 Figure 5.10 A wheel considered as fan Fci ..................... 69 Figure 6.1 A Halin graph H with non-consecutive edges e and f at node x . 74 Figure 6.2 A fan F with centre w ......................... 77 Figure 6.3 A subgraph of H which becomes a fan . 79 Figure 6.4 A tour τ in a wheel, which skips edge ci . 86 Figure 6.5 A wheel considered as fan Fci ..................... 86 Figure 7.1 Example showing the bound in Theorem 7.2.1 is tight . 91 Figure 7.2 Example showing that the bound in Theorem 7.2.3 is tight . 93 x Chapter 1 The quadratic travelling salesman problem 1.1 Introduction E Given a finite set E, a collection F of subsets of E, and a real-valued function f : 2 → R, a combinatorial optimization problem is formulated as Minimize f(S) Subject to S ∈ F. F is often represented in a compact form, without explicitly listing its elements. An element of F is referred to as a feasible solution to the problem and F is the feasible set.