## Geometric facility location under continuous motion: Bounded-velocity approximations to the mobile Euclidean k-centre and k-median problems

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##### Abstract

The traditional problems of facility location are defined statically; a set (or multiset) of n points is given as input, corresponding to the positions of clients, and a solution is returned consisting of set of k points, corresponding to the positions of facilities, that optimizes some objective function of the input set. In the k-centre problem, the objective is to select k points for locating facilities such that the maximum distance from any client to its nearest facility is mini¬mized. In the k-median problem, the objective is to select k points for locating facilities such that the average distance from each client to its nearest facility is minimized. A common setting for these problems is to model clients and facilities as points in Euclidean space and to measure distances between these by the Euclidean distance metric. In this thesis, we examine these problems in the mobile setting. A problem instance consists of a set of mobile clients, each following a continuous trajectory through Euclidean space under bounded velocity. The positions of the mobile Euclidean k-centre and k-median are defined as functions of the instantaneous positions of the clients. Since mobile facilities located at the exact Euclidean k-centre or k-median involve either unbounded velocity or discontinuous motion, we explore approximations to these. The goal is to define a set of functions, corresponding to positions for the set of mobile facilities, that provide a good approximation to the Euclidean k-centre or k-median while maintaining motion that is continuous and whose magnitude of velocity has a low fixed upper bound. Thus, the fitness of a mobile facility is determined not only by the quality of its optimization of the objective function but also by the maximum velocity and continuity of its motion. These additional constraints lead to a trade-off between velocity and approximation factor, requiring new approximation strategies quite different from previous static approximations. We identify existing functions and introduce new functions that provide bounded-velocity approximations of the mobile Euclidean 1-centre, 2-centre, and 1-median. We show that no bounded-velocity approximation of the Eu¬clidean 3-centre or the Euclidean 2-median is possible. Finally, we present kinetic algorithms for maintaining these various functions using both exact and approximate solutions.