Hardness Results and Approximation Algorithms for the Minimum Dominating Tree Problem
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 subtree 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 subtree 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.04498
 Bibcode:
 2018arXiv180204498K
 Keywords:

 Computer Science  Computational Complexity