Optimal dynamic program for r-domination problems over tree decompositions

TL;DR

This paper develops optimal dynamic programming algorithms for r-domination problems on graphs with bounded treewidth, establishing tight bounds under the Strong Exponential Time Hypothesis.

Contribution

It introduces the first algorithms with tight exponential dependence on treewidth for r-domination and connected r-domination problems, extending prior work on NP-hard problems.

Findings

Algorithms run in optimal time under SETH

Dependence on r and treewidth is proven to be tight

Connectivity constraints require an additional exponential factor

Abstract

There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems are optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to $r$-domination problems. In $r$-dominating set, one wished to find a minimum subset $S$ of vertices such that every vertex of $G$ is within $r$ hops of some vertex in $S$. In connected $r$-dominating set, one additionally requires that the set induces a connected subgraph of $G$. We give a $O((2r+1)^{\mathrm{tw}} n)$ time algorithm for $r$-dominating set and a $O((2r+2)^{\mathrm{tw}} n^{O(1)})$ time algorithm for connected $r$-dominating set in $n$-vertex graphs of treewidth $\mathrm{tw}$. We show that the running time dependence on $r$ and $\mathrm{tw}$ is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.