Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances

The distance-d variants of Independent Set and Dominating Set problems have been extensively studied from different algorithmic viewpoints. In particular, the complexity of these problems are well understood on bounded-treewidth graphs [Katsikarelis, Lampis, and Paschos, Discret. Appl. Math 2022][Borradaile and Le, IPEC 2016]: given a tree decomposition of width t, the two problems can be solved in time $d^t \cdot n^{O(1)}$ and $(2d + 1)t \cdot n^{O(1)}$, respectively. Furthermore, assuming the Strong Exponential-Time Hypothesis (SETH), the base constants are best possible in these running times: they cannot be improved to $d-{\epsilon}$ and $2d+1-{\epsilon}$, respectively, for any ${\epsilon} > 0$. We investigate continuous versions of these problems in a setting introduced by Megiddo and Tamir [SICOMP 1983], where every edge is modeled by a unit-length interval of points. In the {\delta}-Dispersion problem, the task is to find a maximum number of points (possibly inside edges) that are pairwise at distance at least {\delta} from each other. Similarly, in the {\delta}-Covering problem, the task is to find a minimum number of points (possibly inside edges) such that every point of the graph (including those inside edges) is at distance at most {\delta} from the selected point set. We provide a comprehensive understanding of these two problems on bounded-treewidth graphs.