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 problems (in-)approximability and offer improvements and extensions of known results f
or Independent Set, of which the problem is a generalization. Specifically, we show: - A lower bound of $Delta^{lfloor d/2rfloor-epsilon}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $Delta$ and an improved upper bound of $O(Delta^{lfloor d/2rfloor})$ on the approximation ratio of any greedy scheme for this problem. - A polynomial-time $2sqrt{n}$-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) $2^{frac{n^{1-epsilon}}{rho d}}$ for even $d$ and $2^{frac{n^{1-epsilon}}{rho(d+rho)}}$ for odd $d$ (under the randomized ETH), complemented by $rho$-approximation algorithms of running times that (almost) match these bounds.
In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of
this problem with respect to various standard graph parameters. In particular, we show the following: - For any $dge2$, an $O^*(d^{textrm{tw}})$-time algorithm, where $textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithms performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(textrm{td}^2)}$-time algorithm parameterized by tree-depth ($textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $epsilon > 0$, runs in time $O^*((textrm{tw}/epsilon)^{O(textrm{tw})})$ and returns a $d/(1+epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists.
In $(k,r)$-Center we are given a (possibly edge-weighted) graph and are asked to select at most $k$ vertices (centers), so that all other vertices are at distance at most $r$ from a center. In this paper we provide a number of tight fine-grained boun
ds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any $rge 1$, we show an algorithm that solves the problem in $O^*((3r+1)^{textrm{cw}})$ time, where $textrm{cw}$ is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithms performance. As a corollary, for $r=1$, this closes the gap that previously existed on the complexity of Dominating Set parameterized by $textrm{cw}$. - We strengthen previously known FPT lower bounds, by showing that $(k,r)$-Center is W[1]-hard parameterized by the input graphs vertex cover (if edge weights are allowed), or feedback vertex set, even if $k$ is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. - We show that the complexity of the problem parameterized by tree-depth is $2^{Theta(textrm{td}^2)}$ by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of $k,r$. In particular, we give algorithms which, for any $epsilon>0$, run in time $O^*((textrm{tw}/epsilon)^{O(textrm{tw})})$, $O^*((textrm{cw}/epsilon)^{O(textrm{cw})})$ and return a $(k,(1+epsilon)r)$-center, if a $(k,r)$-center exists, thus circumventing the problems W-hardness.
