Structurally Parameterized d-Scattered Set

Abstract

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 \(d\ge 2\), an \(O^*(d^{\text {tw}})\)-time algorithm, where \(\text {tw}\) is the treewidth of the input graph and a tight SETH-based lower bound matching this algorithm’s 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 \(2^{O(\text {td}^2)}\)-time algorithm parameterized by tree-depth (\(\text {td}\)), as well as a matching ETH-based lower bound, both for unweighted graphs.