Approximating $\delta$-Dispersion

We consider a continuous facility location problem called $\delta$-Dispersion. For some fixed $\delta > 0$, the goal is to place as many facilities on a graph as possible with pairwise distance at least $\delta$. The facilities may be located on the vertices of the graph, or the interior of the edges. This problem can be interpreted as a continuous version of the well-known Independent Set problem. Its approximation behavior is very similar for large values of $\delta$. Notably, Grigoriev et al. [Algorithmica 21] showed that $\delta$-Dispersion is solvable in polynomial time when $\delta = 1/x$ or $\delta = 2/x$ for a natural number $x$ and NP-hard otherwise. We study the approximability of $\delta$-Dispersion depending on the value of $\delta$. For $\delta > 2$, we show poly-APX-hardness, while for all $\delta < 2$ that are not solvable in polynomial time we show APX-hardness. Thanks to a translation theorem for $\delta$ due to Hartmann et al. [MFCS 22], we may focus our attention for approximation algorithms on the intervals $(2/3 , 1)$ and $(1, 2)$. We provide several approximation algorithms with an approximation factor approaching $1$ as $\delta$ approaches one of the interval boundaries. Surprisingly, the behavior as $\delta$ approaches $2/3$ from above is different: As our hardness reductions reveal, it is impossible (under standard complexity-theoretic assumptions) to construct an approximation algorithm with an approximation factor approaching $1$ as $\delta$ approaches $2/3$ from above.