On the Stochastic Travelling Salesman problem for the Dubin's vehicle

Abstract

In this thesis, I solve the following problem: Given a rectangular region R in which n (n is large) targets are distributed according to some continuous or piece-wise continuous distribution, find the length of the optimal Stochastic Travelling Salesperson tour of a Dubin vehicle over the n targets and design an algorithm that performs within a. constant factor of the optimal expected tour length. We first solve the problem for the case when the distribution of the targets in uniform in R, and then generalize the results to any distribution. To solve the problem, we use an already known lower bound on the expected length of the optimal tour, and we design an algorithm that performs within a. constant factor of that lower bound. To create the constant factor algorithm, we first study the dynamic constraints on the Dubin vehicle to create a building block. and then solve an important auxiliary problem in which targets are not allowed to be too close to each other. After creating the algorithm for the uniform distribution scenario, we establish a lower bound for the scenario where the targets are sampled in R according to a continuous or a piece-wise continuous distribution. We finally generalize our algorithm to the non-uniform scenario and prove that it still performs within a constant factor of the lower bound we proved.