Abstract
The minimum dominating set problem (MDSP) aims to construct the minimum-size subset \(D \subset V\) of a graph \(G = (V, E)\) such that every vertex has at least one neighbor in D. The problem is proved to be NP-hard [5]. In a recent industrial application, we encountered a more general variant of MDSP that extends the neighborhood relationship as follows: a vertex is a k-neighbor of another if there exists a linking path through no more than k edges between them. This problem is called the minimum k-dominating set problem (MkDSP) and the dominating set is denoted as \(D_k\). The MkDSP can be used to model applications in social networks [2] and design of wireless sensor networks [3]. In our case, a telecommunication company uses the problem model to supervise a large social network up to 17 millions nodes via a dominating subset in which k is set to 3.
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