Efficient sub-5 approximations for minimum dominating sets in unit disk graphs

TL;DR

This paper presents three efficient algorithms with approximation ratios of 44/9, 43/9, and 5 for finding minimum dominating sets in unit disk graphs, improving computational efficiency and approximation quality.

Contribution

It introduces new linear-time and near-linear-time algorithms with better approximation ratios for the NP-hard minimum dominating set problem in unit disk graphs.

Findings

Achieved a 44/9-approximation in linear time O(n+m).

Developed a 43/9-approximation algorithm in O(n^2 m) time.

Provided a 5-approximation algorithm with linear time complexity.

Abstract

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTAS. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce a linear-time O(n+m) approximation algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O(n log n) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O(n^2 m) time given only the graph's adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.