Deterministic algorithm for 1-median 1-center two-objective optimization problem

Abstract

K-median and k-center are two well-known problems in facility location which play an important role in operation research, management science, clustering and computational geometry. To the best of our knowledge, although these problems have lots of applications, they have never been studied together simultaneously as a multi objective optimization problem. Multi-objective optimization has been applied in many fields of science where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. In this paper we consider 1-median and 1-center two-objective optimization problem. We prove that Ω(n log n) is a lower bound for proposed problem in one and two dimensions in Manhattan metric. Also, by using the properties of farthest point Voronoi diagram, we present a deterministic algorithm which output the Pareto Front and Pareto Optimal Solutions in O(n log n) time.