A greedy algorithm for the minimum $2$-connected $m$-fold dominating set problem

Abstract

To save energy and alleviate interference in a wireless sensor network, connected dominating set (CDS) has been proposed as the virtual backbone. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a $k$-connected $m$-fold dominating set $((k,m)$-CDS for short): a subset of nodes $C\subseteq V(G)$ is a $(k,m)$-CDS of $G$ if every node in $V(G)\setminus C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$-connected.In this paper, we present an approximation algorithm for the minimum $(2,m)$-CDS problem with $m\ge 2$. Based on a $(1,m)$-CDS, the algorithm greedily merges blocks until the connectivity is raised to two. The most difficult problem in the analysis is that the potential function used in the greedy algorithm is not submodular. By proving that an optimal solution has a specific decomposition, we managed to prove that the approximation ratio is $\alpha +2(1+\ln \alpha )$, where $\alpha$ is the approximation ratio for the minimum $(1,m)$-CDS problem. This improves on previous approximation ratios for the minimum $(2,m)$-CDS problem, both in general graphs and in unit disk graphs.

Appendix

In the appendix, we point out the flaw in the analysis for the $(2,m)$-CDS algorithm on UDG presented in Shang et al. (2008). The algorithm is divided into three steps.

Step 1. Find a maximal independent set $I_0$. Connect $I_0$ by adding a set of connectors $C$. Let $C_{1,1}=I_0\cup C$ be the resulting $(1,1)$-CDS.

Step 2. For $j=1,\ldots ,m-1$, recursively find a maximal independent set $I_j$ in $G-(C_{1,1}\cup I_1\cup \ldots \cup I_{j-1})$. Let $C_{1,m}=C_{1,1}\cup I_1\cup \ldots \cup I_{m-1}$. Then $C_{1,m}$ is a $(1,m)$-CDS with $|C_{1,m}|\le \alpha _m opt$, where $\alpha _m=5+5/m$ for $m\le 5$ and $7$ for $m>5$.

Step 3. Under the assumption that $m\ge 2$, for any $(1,m)$-CDS, there exists a block-bridge with at most two inner nodes the addition of which strictly reduces the number of blocks. By recursively adding such block-bridges, one finally obtains a $(2,m)$-CDS $D$.

Clearly, the number of blocks in $C_{1,m}$ is at most $|C_{1,m}|-1$. Hence $|D|\le |C_{1,m}|+2(|C_{1,m}|-1)<3|C_{1,m}|$. Since $|C_{1,m}|\le \alpha _m opt$, we have $|D|\le 3\alpha _m$, which is $15+\frac{15}{m}$ for $2\le m\le 5$ and $21$ for $m>5$.

The flaw in Shang et al. (2008) is as follows. It was proved that Step 2 does not increase the number of cut vertices. Hence the authors use $|D|\le |C_{1,m}|+2(|C_{1,1}|-1)$ to get their claimed approximation ratio. The problem is: Step 2 does not increase the number of cut vertices, but adding block-bridge in Step 3 might not strictly reduce the number of cut vertices (this can be seen by considering a star); Step 3 strictly reduces the number of blocks, but the number of blocks in $C_{1,m}$ might be larger than that in $C_{1,1}$.