Dispersing Obnoxious Facilities on Graphs by Rounding Distances

We continue the study of $\delta$-dispersion, a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that every two facilities have distance at least $\delta$ from each other. Our main technical contribution is an efficient procedure to `round-up' distance $\delta$. It transforms a $\delta$-dispersed set $S$ into a $\delta^\star$-dispersed set $S^\star$ of same size where distance $\delta^\star$ is a slightly larger rational $\tfrac{a}{b}$ with a numerator $a$ upper bounded by the longest (not-induced) path in the input graph. Based on this rounding procedure and connections to the distance-$d$ independent set problem we derive a number of algorithmic results. When parameterized by treewidth, the problem is in XP. When parameterized by treedepth the problem is FPT and has a matching lower bound on its time complexity under ETH. Moreover, we can also settle the parameterized complexity with the solution size as parameter using our rounding technique: $\delta$-\dispersion is FPT for every $\delta \leq 2$ and W[1]-hard for every $\delta > 2$. Further, we show that $\delta$-dispersion is NP-complete for every fixed irrational distance $\delta$, which was left open in a previous work.