The Maximum Facility Location Problem

In the maximum facility location problem, we are given a set of clients C and facilities F . Assigning a client u to a facility v incurs a cost w uv and each facility has an associated interval I v on the real line. Our goal is to assign as many clients to open facilities in order to maximise the cost of assignment, subject to the constraint that open facilities cannot have their respective intervals overlap. In this thesis we design two new approximation algorithms for this problem that use LProunding techniques. The first is 0.19-approximation algorithm, and the second is a randomised rounding algorithm. Afterwards, we turn our attention of the hardness of the problem, and show that the maximum facility location problem cannot be approximated in polynomial time within a factor of (1 1/e+ ") for some " > 0 unless P = NP. In order to demonstrate the utility of our algorithm, we apply it to a biological scenario in which we aim to map sequences of nucleotides (reads) from a given donor genome to a reference genome. We show that our algorithms are up to two order of magnitudes faster than existing algorithms, without a sacrifice in the quality of results. iii Acknowledgements First and foremost, I would like to thank Dr. Julián Mestre for his expert knowledge on the subject, continued support and encouragement throughout this project. I’d like to thank both Dr. Stefan Canzar and Dr. Khaled Elbassioni for their help and contributions to this project. In particular Dr. Stefan Canzar, for his great explanations in helping me understand the biological nature of the problem, providing me with data sets (produced by CLEVER) and help interpreting the results. iv CONTENTS Student Plagiarism: Compliance Statement ii Abstract iii Acknowledgements iv List of Figures vii List of Tables ix Notation x Chapter