Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams

We study a general family of facility location problems defined on planar graphs and on the 2-dimensional plane. In these problems, a subset of $k$ objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the $n^{O(k)}$ time brute force algorithm of selecting $k$ objects can be improved to $n^{O(\sqrt{k})}$ time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Voronoi diagram of a hypothetical solution of $k$ objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then recurse on the resulting subproblems. We complement our study by giving evidence that packing problems have $n^{O(\sqrt{k})}$ time algorithms for a much more general class of objects than covering problems have.