Efficient Approximations for the Online Dispersion Problem

Abstract

The dispersion problem has been widely studied in computational geometry and facility location and is closely related to the packing problem. The goal is to locate $n$ points (e.g., facilities or persons) in a $k$-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios, however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore, we study, for the first time in the literature, the online dispersion problem in Euclidean space. There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly nontrivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance. For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is $(2\ln 2+\epsilon)$-competitive, where $\epsilon>0$ can be arbitrarily small and the algorithm's running time is polynomial in $\frac{1}{\epsilon}$. We show that this algorithm is actually optimal. For the same problem in a square, we provide a $1.591$-competitive algorithm and a $1.183$ lower bound. Furthermore, for arbitrary $k$-dimensional polytopes with $k\geq 2$, we provide a $\frac{2}{1-\epsilon}$-competitive algorithm and a $\frac{7}{6}$ lower bound. All our lower bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary $k$-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions.