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

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Summary

Introduction

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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