Performance Guaranteed Approximation Algorithm for Minimum $k$-Connected $m$-Fold Dominating Set

TL;DR

This paper introduces a new approximation algorithm for constructing fault-tolerant virtual backbones in wireless sensor networks, guaranteeing performance for general k and m, improving upon previous limitations.

Contribution

It presents the first approximation algorithm with guaranteed performance for general k and m in $(k,m)$-connected dominating sets, extending beyond prior work limited to small k.

Findings

Achieves a performance ratio of $(2k-1) imes ext{performance ratio of minimum CDS}$

Performance ratio is $O( ext{ln} riangle)$, where $ riangle$ is maximum degree

Extends approximation guarantees to all $k eq 3$, $m eq k$ cases

Abstract

To achieve an efficient routing in a wireless sensor network, connected dominating set (CDS) is used as virtual backbone. A fault-tolerant virtual backbone can be modeled as a $(k,m)$-CDS. For a connected graph $G=(V,E)$ and two fixed integers $k$ and $m$, a node set $C\subseteq V$ is a $(k,m)$-CDS of $G$ if every node in $V\setminus C$ has at least $m$ neighbors in $C$, and the subgraph of $G$ induced by $C$ is $k$-connected. Previous to this work, approximation algorithms with guaranteed performance ratio in a general graph were know only for $k\leq 3$. This paper makes a significant progress by presenting a $(2k-1)\alpha_0$ approximation algorithm for general $k$ and $m$ with $m\geq k$, where $\alpha_0$ is the performance ratio for the minimum CDS problem. Using currently best known ratio for $\alpha_0$, our algorithm has performance ratio $O(\ln\Delta)$, where $\Delta$ is the maximum degree of the graph.