## Algorithms for Solving Variations of the Traveling Salesman Problem

Baniasadi, Pouya, 2019 Algorithms for Solving Variations of the Traveling Salesman Problem, Flinders University, College of Science and Engineering

### Abstract

The Traveling Salesman Problem (TSP) is widely considered to be one of the most important problems in optimization and computer science. The reasons behind the significance of TSP are both its theoretical connections to complexity theory and algorithmic optimization, and its potential applications in various fields such as logistics and genetics. In this thesis, I consider three important variations of TSP from the standpoint of algorithmic solutions and applications. These variations are the Hamiltonian Cycle Problem (HCP), the Sparse Traveling Salesman Problem (STSP) and the Clustered Generalized Traveling Salesman Problem (CGTSP).

Given a graph, the HCP is the problem of finding a simple cycle that includes all nodes of the graph, and it is equivalent to the TSP where the weights are binary values. Despite its deceptively simple definition, there is no known efficient algorithm for solving the HCP as it belongs to the important class of NP-complete problems. As a special and difficult subset of the TSP, the study of HCP is a potentially rewarding direction of research for understanding complexities involved in the TSP itself. In Chapter 2, I analyze and further develop a recently introduced approach for solving the HCP called the Snakes and Ladders Heuristic (SLH). SLH is demonstrated to be on par with state-of-the-art approaches for solving HCP,

and in some instances, it outperforms other approaches. An abstract discussion of the algorithmic principles at play in SLH is included, and accordingly, an improved algorithmic approach for HCP is proposed.

Sparse Traveling Salesman Problem (STSP) is a slight variation of the TSP on sparse instances. The primary motivation for considering this special case of TSP is its algorithmic significance. Most successful algorithmic approaches for solving TSP involves the initial sparsification of the instance, suggesting that much of the computational complexity of TSP comes from STSP. In Chapter 3, I discuss the creation of meaningful benchmark instances of STSP that provide insightful information about the performance of a TSP algorithm. Using these instances, I report on a computational study, involving over eight years of CPU time, and examine some of the state-of-the-art TSP algorithms. Next, I use the insights gained from the computational study to identify a set of effective algorithmic approaches for solving STSP in Chapter 4. I implement and combine these approaches to produce effective hybrid algorithms for the STSP. Finally, I explore the applications of STSP algorithms by formulating two well-known problems in terms of an STSP. These two problems are the Time-Dependant Traveling Salesman Problem and the DNA-assembly problem.

In Chapter 5, I describe an extension of the TSP called CGTSP. From the perspective of real-world applications, a CGTSP model provides much more flexibility for accommodating a wider range of assumptions compared to a classical TSP model. I develop a transformation method for solving CGTSP and demonstrate substantial improvements to the existing solutions methods for CGTSP. Finally, I then illustrate the potential applications of the CGTSP in two modern problems in logistics, namely the robotic automated storage and retrieval system and the drone-assisted delivery systems. Finally, I report a comprehensive computational study on the effectiveness of the transformation method.

Keywords: traveling salesman problem, TSP, variations of TSP, Hamiltonian cycle problem, combinatorial optimization, operations research, logistics

Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2019
School: College of Science and Engineering