Greedy approximation for the minimum connected dominating set with labeling

Abstract

Given a connected graph $G=(V,E)$. A subset $C\subseteq V$ is a dominating set if every vertex of V is either in C or adjacent to a vertex in C. Further, C is a connected dominating set if C is a dominating set and the induced subgraph G[C] is connected. The Minimum Connected Dominating Set (Min-CDS) problem asks to find a connected dominating set with the minimum size, which finds applications in communication networks, in particular, as a virtual backbone in wireless sensor networks. This paper focuses on a variant of the classic Min-CDS problem, called Minimum Connected Dominating Set with Labeling (Min-CDSL), in which we are given a connected graph with vertex labels, and are required to find a connected dominating set C such that the number of labels in C (instead of |C|) is minimized. Min-CDSL is apparently a generalization of Min-CDS, and is undoubtedly $\mathcal {NP}$-$\mathrm {complete}$. We give an approximation algorithm for Min-CDSL within performance ratio bounded by $\ln |V(G)|+\mathrm {span}(G)+1$, where $\mathrm {span}(G)$ refers to the maximum span of the input labeled graph (i.e., the number of connected components of the induced subgraph by a single label). In general, $\mathrm {span}(G)\ll |V(G)|$ and for a series of labeled graphs $\mathrm {span}(G)=O(1)$. For a random graph $G\in {G_{n,p}}$, $\mathrm {span}(G)=O(\ln |V(G)|)$ almost surely, and thus our approximation ratio is $O(\ln |V(G)|)$ which is reasonable comparing with the best known approximation ratio $\ln |V(G)|+1$ for Min-CDS.