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 $d-ε$ and $2d+1-ε$, respectively, for any $ε > 0$. 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 δ-Dispersion problem, the task is to find a maximum number of points (possibly inside edges) that are pairwise at distance at least δ from each other. Similarly, in the δ-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 δ from the selected point set. We provide a comprehensive understanding of these two problems on bounded-treewidth graphs.

翻译：独立集和支配集问题的距离-d变体已从不同算法视角得到广泛研究。特别地，这些问题在有界树宽图上的复杂度已被充分理解[Katsikarelis, Lampis, and Paschos, Discret. Appl. Math 2022][Borradaile and Le, IPEC 2016]：给定宽度为t的树分解，两个问题可分别在时间$d^t \cdot n^{O(1)}$和$(2d + 1)t \cdot n^{O(1)}$内求解。此外，在强指数时间假设(SETH)下，这些运行时间中的底数常数是最优的：对于任何$ε > 0$，它们分别无法改进为$d-ε$和$2d+1-ε$。我们研究了Megiddo和Tamir [SICOMP 1983]所引入设定中这些问题的连续版本，其中每条边被建模为单位长度的点区间。在δ-分散问题中，任务是找到两两之间距离至少为δ的（可能位于边内部的）最大点数。类似地，在δ-覆盖问题中，任务是找到最小数量的（可能位于边内部的）点，使得图中每个点（包括边内部的点）到所选点集的距离至多为δ。我们对有界树宽图上的这两个问题提供了全面的理解。