Online Facility Location with Mobile Facilities

We examine the Online Facility Location Problem in an augmented version, where the online algorithm is allowed to adapt the position of the facilities for costs proportional to the distance by which the position is changed. In this setting, it is possible to construct online algorithms which deal with the lower bound instances of Online Facility Location much more effectively. Fotakis showed a lower bound of $Ømega(\fracłog n łog łog n )$ for the original Online Facility Location Problem, where n denotes the number clients. This bounds holds even on the real line and for randomized algorithms against oblivious adversaries. In contrast, we are able to achieve competitive ratios independent of n in our model. We propose randomized online algorithms in two settings: We consider the Euclidean space (of arbitrary dimension) and allow the facilities to either move arbitrarily or to move at most a constant distance m in each time step. The costs for moving a facility from a to b is $D\cdot d(a,b)$ where $D\geq 1$ is a constant. Our algorithms are memoryless w.r.t. past requests and only make local modifications to at most one facility in each time step. In the case of arbitrary movement, the competitive ratio only depends on D . In the case of limiting the movement to a constant distance m, the competitive ratio additionally depends on the opening cost $c_f$ of facilities and m . We show that our results are asymptotically tight on the real line. For the Euclidean space of higher dimensions, the competitive ratio of our algorithms is tight with respect to D, $c_f$ and m, but is additionally impacted by the number of optimal facilities.