Continuous facility location on graphs

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range [Formula: see text] . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most [Formula: see text] from one of these facilities. We investigate this covering problem in terms of the rational parameter [Formula: see text] . We prove that the problem is polynomially solvable whenever [Formula: see text] is a unit fraction, and that the problem is NP-hard for all non unit fractions [Formula: see text] . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for [Formula: see text] , and it is W[2]-hard for [Formula: see text] .