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.

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解决超大型网络上的k占优集问题

众所周知的最小支配集问题（MDSP）旨在构造图中顶点的最小大小子集，以使每个其他顶点在子集中具有至少一个邻居。在本文中，我们研究了扩展邻域关系的问题的一般版本：如果两个顶点之间存在不超过k个边的路径，则两个顶点称为彼此的邻居。如果k为1，则称为“最小k支配集问题”（MkDSP）的问题成为经典的支配集问题，并且在监视大型社交网络中具有重要的应用。我们提出了一种有效的启发式算法，该算法可以处理具有多达1700万个顶点和3300万条边的现实世界实例。这是第一次针对最小的k控制集问题求解如此大的图。