A New Approximation Algorithm for Minimum-Weight $(1,m)$--Connected Dominating Set

Abstract

Consider a graph with nonnegative node weight. A vertex subset is called a CDS (connected dominating set) if every other node has at least one neighbor in the subset and the subset induces a connected subgraph. Furthermore, if every other node has at least $m$ neighbors in the subset, then the node subset is called a $(1,m)$CDS. The minimum-weight $(1,m)$CDS problem aims at finding a $(1,m)$CDS with minimum total node weight. In this paper, we present a new polynomial-time approximation algorithm for this problem with approximation ratio $2H({\delta }_{\max }+m-1)$, where ${\delta }_{\max }$ is the maximum degree of the given graph and $H(\cdot )$ is the Harmonic function, i.e., $H(k)={\sum }_{i=1}^k\frac{1}{i}$.