The Kinetic Facility Location Problem

We present a deterministic kinetic data structure for the facility location problem that maintains a subset of the moving points as facilities such that, at any point of time, the accumulated cost for the whole point set is at most a constant factor larger than the optimal cost. Each point can change its status between client and facility and moves continuously along a known trajectory in a d-dimensional Euclidean space, where dis a constant. Our kinetic data structure requires $\mathcal{O}(n (\log^{d}(n)+\log(nR)))$ space, where $R:=\frac{\max_{p_i \in \mathcal{P}}{f_i} \,\cdot\, \max_{p_i\in \mathcal{P}}{d_i}}{\min_{p_i \in \mathcal{P}}{f_i} \,\cdot\, \min_{p_i\in \mathcal{P}}{d_i}}$, $\mathcal{P} = \{ p_1, p_2, \ldots , p_n \}$ is the set of given points, and f i , d i are the maintenance cost and the demand of a point p i , respectively. In case that each trajectory can be described by a bounded degree polynomial, we process $\mathcal{O}(n^2 \log^2(nR))$ events, each requiring $\mathcal{O}(\log^{d+1}(n) \cdot \log(nR))$ time and $\mathcal{O}(\log(nR))$ status changes. To the best of our knowledge, this is the first kinetic data structure for the facility location problem.