Abstract
The well-known minimum dominating set problem (MDSP) aims to construct the minimum-size subset of vertices in a graph such that every other vertex has at least one neighbor in the subset. In this article, we study a general version of the problem that extends the neighborhood relationship: two vertices are called neighbors of each other if there exists a path through no more than k edges between them. The problem called “minimum k-dominating set problem” (MkDSP) becomes the classical dominating set problem if k is 1 and has important applications in monitoring large-scale social networks. We propose an efficient heuristic algorithm that can handle real-world instances with up to 17 million vertices and 33 million edges. This is the first time such large graphs are solved for the minimum k-dominating set problem.
Highlights
Problem context and definition The well-known minimum dominating set problem (MDSP) deals with determining the smallest dominating set of a given graph G = (V, E)
We take the viewpoint of a company that runs a very large social network in which users can be modeled as nodes and the relationship among users can be modeled as edges
We need to consider the general version of dominating set named k-dominating set Dk which is defined as following: each vertex either belongs to the Dk or is connected to at least one member of Dk through a path of no more than k edges
Summary
Problem context and definition The well-known minimum dominating set problem (MDSP) deals with determining the smallest dominating set of a given graph G = (V , E). We take the viewpoint of a company that runs a very large social network in which users can be modeled as nodes and the relationship among users can be modeled as edges. A potential solution is to construct a subset of users that can represent key properties of the network. In the case of social network scale, it is still too expensive to construct a dominating set because the size of Nguyen et al Comput Soc Netw (2020) 7:4 the dominating set could be large. The classical minimum dominating set corresponds to a special case when k = 1. For value k > 1 , the cardinality of k-dominating set is less than that of 1-dominating set:|Dk | ≤ |D1| , the monitoring cost of the network is reduced
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