Approximating $\delta$-Covering

$\delta$-Covering, for some covering range $\delta>0$, is a continuous facility location problem on undirected graphs where all edges have unit length. The facilities may be positioned on the vertices as well as on the interior of the edges. The goal is to position as few facilities as possible such that every point on every edge has distance at most $\delta$ to one of these facilities. For large $\delta$, the problem is similar to dominating set, which is hard to approximate, while for small $\delta$, say close to $1$, the problem is similar to vertex cover. In fact, as shown by Hartmann et al. [Math. Program. 22], $\delta$-Covering for all unit-fractions $\delta$ is polynomial time solvable, while for all other values of $\delta$ the problem is NP-hard. We study the approximability of $\delta$-Covering for every covering range $\delta>0$. For $\delta \geq 3/2$, the problem is log-APX-hard, and allows an $\mathcal O(\log n)$ approximation. For every $\delta < 3/2$, there is a constant factor approximation of a minimum $\delta$-cover (and the problem is APX-hard when $\delta$ is not a unit-fraction). We further study the dependency of the approximation ratio on the covering range $\delta < 3/2$. By providing several polynomial time approximation algorithms and lower bounds under the Unique Games Conjecture, we narrow the possible approximation ratio, especially for $\delta$ close to the polynomial time solvable cases.