Hardness Results and Approximation Algorithms for the Minimum Dominating Tree Problem

TL;DR

This paper establishes the equivalence in approximability between the Minimum Dominating Tree problem and the Group Steiner Tree problem, improving known approximation bounds and exploring related variants like dominating star and path.

Contribution

It proves the approximation equivalence of the Minimum Dominating Tree and Group Steiner Tree problems, advancing the understanding of their computational complexity.

Findings

Equivalence in approximability between the two problems.

Improved approximation algorithms and inapproximability bounds.

Analysis of variants like dominating star and path.

Abstract

Given an undirected graph $G = (V, E)$ and a weight function $w:E \to \mathbb{R}$, the \textsc{Minimum Dominating Tree} problem asks to find a minimum weight sub-tree of $G$, $T = (U, F)$, such that every $v \in V \setminus U$ is adjacent to at least one vertex in $U$. The special case when the weight function is uniform is known as the \textsc{Minimum Connected Dominating Set} problem. Given an undirected graph $G = (V, E)$ with some subsets of vertices called groups, and a weight function $w:E \to \mathbb{R}$, the \textsc{Group Steiner Tree} problem is to find a minimum weight sub-tree of $G$ which contains at least one vertex from each group. In this paper we show that the two problems are equivalents from approximability perspective. This improves upon both the best known approximation algorithm and the best inapproximability result for the \textsc{Minimum Dominating Tree} problem. We also consider two extrema variants of the \textsc{Minimum Dominating Tree} problem, namely, the \textsc{Minimum Dominating Star} and the \textsc{Minimum Dominating Path} problems which ask to find a minimum dominating star and path respectively.