Approximation Algorithms for Steiner Connected Dominating Set

Abstract

Steiner connected dominating set (SCDS) is a generalization of the famous connected dominating set problem, where only a specified set of required vertices has to be dominated by a connected dominating set, and known to be NP-hard. This paper firstly modifies the SCDS algorithm of Guha and Khuller and achieves a worst case approximation ratio of (2+1/(m−1))H(min (D, k))+O(1), which outperforms the previous best result (c+1)H(min (D, k))+O(1) in the case of mge 1+1/(c−1), where c is the best approximation ratio for Steiner tree, D is the maximum degree of the graph, k is the cardinality of the set of required vertices, m is an optional integer satisfying 0≤ m ≤ min (D, k) and H is the harmonic function. This paper also proposes another approximation algorithm which is based on a greedy approach. The second algorithm can establish a worst case approximation ratio of 2ln (min (D, k))+O(1), which can also be improved to 2ln k if the optimal solution is greater than $\frac{c\dot{e^{2c+1}}}{2(c+1)}$.