A greedy approximation for minimum connected dominating sets

Lu Ruan, Hongwei Du, Xiaohua Jia, Weili Wu, Yingshu Li, Ker-I Ko

Abstract: Given a graph, a connected dominating set is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected. A minimum connected dominating set is such a vertex subset with minimum cardinality. In this paper, we present a new one-step greedy approximation with performance ratio lnδ+2<math><mi is="true">ln</mi><mi is="true">δ</mi><mo is="true">+</mo><mn is="true">2</mn></math> where δ<math><mi is="true">δ</mi></math> is the maximum degree in the input graph. The interesting aspect is that the greedy potential function of this algorithm is not supmodular while all previously known one-step greedy algorithms with similar performance have supmodular potential functions.