Approximation Algorithm for Minimum Weight Connected $m$-Fold Dominating Set

TL;DR

This paper presents an approximation algorithm for the minimum weight (1,m)-connected dominating set problem, improving fault-tolerance in wireless network backbones with a focus on unit disk graphs.

Contribution

It introduces a new approximation algorithm with bounds depending on maximum degree, enabling better fault-tolerant backbone design in wireless networks.

Findings

Provides an approximation ratio of (H(δ+m)+2H(δ-1)) for the problem.

Achieves a (6.67+ε)-approximation for unit disk graphs.

Extends previous work by replacing n with δ in approximation bounds.

Abstract

Using connected dominating set (CDS) to serve as a virtual backbone in a wireless networks can save energy and reduce interference. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. A $k$-connected $m$-fold dominating set ($(k,m)$-CDS) of a graph $G$ is a node set $D$ such that every node in $V\setminus D$ has at least $m$ neighbors in $D$ and the subgraph of $G$ induced by $D$ is $k$-connected. Using $(k,m)$-CDS can tolerate the failure of $\min\{k-1,m-1\}$ nodes. In this paper, we study Minimum Weight $(1,m)$-CDS problem ($(1,m)$-MWCDS), and present an $(H(\delta+m)+2H(\delta-1))$-approximation algorithm, where $\delta$ is the maximum degree of the graph and $H(\cdot)$ is the Harmonic number. Notice that there is a $1.35\ln n$-approximation algorithm for the $(1,1)$-MWCDS problem, where $n$ is the number of nodes in the graph. Though our constant in $O(\ln \cdot)$ is larger than 1.35, $n$ is replaced by $\delta$. Such a replacement enables us to obtain a $(6.67+\varepsilon)$-approximation for the $(1,m)$-MWCDS problem on unit disk graphs.