Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center

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 bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any r≥ 1, we show an algorithm that solves the problem in O^*((3r+1)^textrmcw) time, where textrmcw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by textrmcw. - We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph's 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(textrmtd2) 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 ε>0, run in time O^*((textrmtw/ε)^O(textrmtw)), O^*((textrmcw/ε)^O(textrmcw)) and return a (k,(1+ε)r)-center, if a (k,r)-center exists, thus circumventing the problem's W-hardness.