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

2026-06-03

AI Summary

This work addresses the δ-Dispersion (continuous independent set) and δ-Covering (continuous dominating set) problems on graphs of bounded treewidth, where vertices may be placed at arbitrary real-valued positions along edges and distances are measured in the continuous metric. Leveraging tree decompositions, the authors design exact dynamic programming algorithms that carefully model state transitions under real-valued distances, achieving running times of \(d^t \cdot n^{O(1)}\) for δ-Dispersion and \((2d+1)^t \cdot n^{O(1)}\) for δ-Covering, where \(t\) denotes the treewidth. This study provides the first systematic characterization of the computational complexity of these two continuous optimization problems on bounded-treewidth graphs and establishes tight lower bounds on the exponential dependence on treewidth under the Strong Exponential Time Hypothesis (SETH), proving that the obtained bases cannot be improved.

Abstract

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.

Problem

• bounded-treewidth graphs
• δ-Dispersion
• δ-Covering
• continuous distance
• graph algorithms

Innovation

• bounded-treewidth graphs
• continuous dispersion
• distance-constrained covering
• parameterized complexity
• Strong Exponential-Time Hypothesis