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

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 applicability 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 PTASs. 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 an O(n + m) 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 43/9approximation 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.

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

“…Eqs. (25) and (26) implies ϕ n+1 − φ n+1 = ϕ n (ϕ 2 n − φ 2 n ), which together with ϕ 2 = φ 2 = 2 shows that ϕ n = φ n for all n ≥ 2. Then, considering Eq.…”

Efficient independent set approximation in unit disk graphs

“…There are polynomialtime approximation schemes (PTASs) for several optimization problems on unit disk graphs [10,14,17,18,22,23], including maximum (weight) independent set. However, the high complexities of the PTASs motivated the recent study of faster constantapproximation algorithms, notably for minimum dominating set [5,12,13,15] and maximum (weight) independent set [13,14].…”

On Forbidden Induced Subgraphs for Unit Disk Graphs

“…For example, the fact was utilized in [13] for obtaining 3-approximation algorithm for the maximum independent set problem and 5-approximation algorithm for the dominating set problem. In [7] da Fonseca et al used additional geometrical restrictions of UDGs to design an algorithm for the latter problem with better approximation factor 44/9. The authors pointed out that further improvement may require new information about forbidden induced subgraphs for UDGs, and in a subsequent paper [8] they developed algorithm for recognizing UDGs.…”