Improved (In-)Approximability Bounds for d-Scattered Set

In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem’s (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. Specifically, we show:

• A lower bound of ${\mathrm{\Delta }}^{\lfloor d\mathrm{/}2\rfloor -ϵ}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $\mathrm{\Delta }$ and an improved upper bound of $O({\mathrm{\Delta }}^{\lfloor d\mathrm{/}2\rfloor })$ on the approximation ratio of any greedy scheme for this problem.
• A polynomial-time $2\sqrt{n}$-approximation for bipartite graphs and even values of $d$, that matches the known lower bound by considering the only remaining case.
• A lower bound on the complexity of any $\rho$-approximation algorithm of (roughly) $2^{\frac{n^{1-ϵ}}{\rho d}}$ for even $d$ and $2^{\frac{n^{1-ϵ}}{\rho (d+\rho )}}$ for odd $d$ (under the randomized ETH), complemented by $\rho$-approximation algorithms of running times that (almost) match these bounds.