A greedy approximation algorithm for the minimum (2,2)-connected dominating set problem

TL;DR

This paper introduces a two-phase greedy algorithm for finding a fault-tolerant (2,2)-connected dominating set in wireless sensor networks, improving the approximation factor over previous methods.

Contribution

It presents a novel greedy algorithm with a better approximation ratio for the (2,2)-connected dominating set problem in virtual networks.

Findings

Achieves an approximation factor of (3+ln(Δ+2))

Improves upon previous performance bounds

Applicable to fault-tolerant wireless sensor network design

Abstract

Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless sensor network (WSN) is an effective way to save energy and reduce the impact of broadcasting storms. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. This could be modeled as a k-connected, m-fold dominating set ((k,m)-CDS). Given a virtual undirected network G=(V,E), a subset C\subset V is a (k,m)-CDS of G if (i) G[C], the subgraph of G induced by C is k-connected, and (ii) each node in V\C has at least m neighbors in C. We present a two-phase greedy algorithm for computing a (2,2)-CDS that achieves an asymptotic approximation factor of $(3+\ln(\Delta+2))$, where $\Delta$ is the maximum degree of G. This result improves on the previous best known performance factor of $(4+\ln\Delta+2\ln(2+\ln\Delta))$ for this problem.

Equations

$$m_C(v)=\{\begin{array}{ll}0,& \text{if }v\in C,\text{or adjacent to at least }2\text{ nodes in C}\\ 1,& \text{if }v\in V\setminus C,\text{ and adjacent to at most }1\text{ node in }C\text{.}\end{array}$$

$$\begin{array}{rl}\hat{p}(C)& =\underset{x\in C}{\max }p(C\setminus \{x\})\\ & =p(C\setminus \{x_C\})\end{array}$$

$$f(C)=\hat{p}(C)+q(C)+m(C)$$

$${\Delta }_{y}f(C)=f(C)-f(C\cup \{y\}),$$

$${\Delta }_{y}f(C)={\Delta }_y\hat{p}(C)+{\Delta }_{y}q(C)+{\Delta }_{y}m(C)$$

$${\Delta }_{y}f(A\cup B)\le {\Delta }_{y}f(A)+1$$

$$\mu (f)=\mu (\hat{p})+\mu (q)+\mu (m).$$

$$\mu(m)=\left\{\begin{array}[c]{l}s-1,~{}\text{if }y\text{ is gray for }A\cup B,\text{ {but} not gray for }A\\ s,\text{ otherwise.}\end{array}\right.$$

$$\mu(q)\leq\left\{\begin{array}[c]{l}-s,~{}\text{if }y\text{ is adjacent to }B\\ -(s-1),\text{ otherwise.}\end{array}\right.$$

$$\hat{p}(A\cup \{y\})=\max \{\alpha -\gamma ,\beta \}+1$$

$$\begin{array}{rl}{\Delta }_y\hat{p}(A)& =\hat{p}(A)-\hat{p}(A\cup \{y\})\\ & =\alpha +\beta -[\max \{\alpha -\gamma ,\beta \}]+1\\ & =\alpha +\beta +\min \{\gamma -\alpha ,-\beta \}-1\\ & =\min \{\beta +\gamma ,\alpha \}-1\end{array}$$

$$\mu(\hat{p})\text{~{}~{}}\left\{\begin{array}[c]{l}=0,\text{ if }y\text{ is not adjacent to }B,\\ \leq 1,\text{ otherwise.}\end{array}\right.$$

$$\begin{array}{rl}f(C_{i-1})-2& =f(C_{i-1})-f(C_{i-1}\cup C^{\ast })\\ & =\sum _{j=1}^{\theta }{\Delta }_{y_j}(C_{i-1}\cup C_{j-1}^{\ast })\\ & \le \sum _{j=1}^{\theta }\left({\Delta }_{y_j}(C_{i-1})+1\right)\end{array}$$

$${\Delta }_{y_j}f(C_{i-1})+1\ge \frac{f(C_{i-1})-2}{\theta }$$

$${\Delta }_{x_i}f(C_{i-1})\ge {\Delta }_{y}f(C_{i-1})\ge \frac{f(C_{i-1})-2}{\theta }-1$$

$$\text{or }f(C_i)\le f(C_{i-1})-\frac{f(C_{i-1})-2}{\theta }+1$$

$$a_i\le a_{i-1}-\frac{a_{i-1}}{\theta }+1$$

$$a_i\le a_{i-1}-\lceil \frac{a_{i-1}}{\theta }\rceil +1$$

$$a_{i}\leq a_{i-1}\left(1-\frac{1}{\theta}\right)+1\text{, whose solution, as in~{}\cite[cite]{[\@@bibref{}{Guha_Khuller_1998,Ruan_Du_Jia_Wu_Li_Ko_2004,Zhou_Zhang_Wu_Xing_2014}{}{}]}, is }$$

$$a_i\le a_0{\left(1-\frac{1}{\theta }\right)}^i+\sum _{j=0}^{i-1}{\left(1-\frac{1}{\theta }\right)}^j$$

$$\theta \ln (a_0/\theta )+\theta +2\theta +4=\theta \left[\ln \frac{a_0}{\theta }+3+\frac{4}{\theta }\right]$$

$${\Delta }_{x_1}f(\varnothing )=|N_G(x_1)|+2=\Delta +2$$

$$f(C_1)\le f(C_0)-\frac{f(C_0)-2}{\theta }+1\text{or}$$

$$\frac{2n-2}{\theta }\le \Delta +2\text{or}$$

$$\frac{2n}{\theta }\le \Delta +2+\frac{2}{\theta }$$

Full text

A greedy approximation algorithm for the minimum $(2,2)$-connected dominating

set problem

Yash P. Aneja

Odette School of Business

University of Windsor

Windsor, Canada

  

Asish Mukhopadhyay

School of Computer Science

University of Windsor

Windsor Canada

  

Md. Zamilur Rahman

School of Computer Science

University of Windsor

Windsor Canada

Abstract Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless sensor network (WSN) is an effective way to save energy and reduce the impact of broadcasting storms. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. This could be modeled as a $k$-connected, $m$-fold dominating set ($(k,m)$-CDS). Given a virtual undirected network $G=(V,E),$ a subset $C\subset V$ is a $(k,m)$-CDS of $G$ if (i) $G[C],$ the subgraph of $G$ induced by $C,$ is $k$-connected, and (ii) each node in $V\backslash C$ has at least $m$ neighbors in $C.$ We present a two-phase greedy algorithm for computing a $(2,2)$-CDS that achieves an asymptotic approximation factor of $(3+\ln(\Delta+2)),$ where $\Delta$ is the maximum degree of $G.$ This result improves on the previous best known performance factor of $(4+\ln\Delta+2\ln(2+\ln\Delta))$ for this problem.

1 Introduction

Suppose $G=(V,E)$ is a connected graph. A subset $C$ of $V$ is a said to be a connected dominating set (CDS) of $G$ if $G[C]$, the induced graph on $C$, is connected and every vertex $v$ in $V\setminus C$ is a neighbor of $C$ (connected by an edge to some vertex $u\in C$). Nodes in $C$ are called dominators, and nodes in $V\backslash C$ are called dominatees. To save energy and reduce interference, it is desirable that the CDS size is as small as possible. Computing a minimum CDS is a well known NP-hard problem [3]. By showing that finding a minimum set cover is a special case of finding a minimum CDS, Guha and Khullar [4] established that a minimum CDS can not be approximated within $\rho\ln n$ for any 0<$$\rho<1 unless $NP\subset DTIME(N^{O(\log\log n)}).$ In the same paper Guha and Khullar [4] proposed a two-phase greedy algorithm, with an approximation factor of $(3+\ln\Delta)$ for fining a minimum sized CDS. Subsequently, Ruan et. al. [5] used a potential function approach to come up with a single phase greedy algorithm improving the approximation ratio to $(2+\ln\Delta).$ There are, in the literature, several approximation algorithms for finding a minimum CDS for a general graph [2].

To make a virtual backbone more robust to deal with frequent node failures in WSNs, researchers have suggested using a $(k,m)$-CDS. As mentioned in the abstract, $C\subset V$ is a $(k,m)$-CDS if every node in $V\backslash C$ is adjacent to at least $m$ nodes in $C,$ and $G[C],$ the subgraph induced by $C,$ is $k$-connected. The $k$-connectedness means that $|C|>k$ and $G[C\backslash X]$ is connected for any $X\subset C$ with $|X|<k$. In other words, no two vertices of $G[C]$ are separated by removal of fewer than $k$ other vertices of $C$. With such a $C,$ messages can be shared by the whole network, where every node in $V\backslash C$ can tolerate up to $m-1$ faults (node failures) on its dominators, and the virtual backbone $G[C]$ can tolerate up to $k-1$ faults.

Zhou et al. [7], using a more complex potential function than the one in Ruan et al. [5], provide a single phase $(2+\ln(\Delta+m-2))$-approximation algorithm for the minimum $(1,m)$-CDS problem in a general graph.

Shi et al. [6], using a two-phase approach, provide a $(\alpha+2(1+\ln\alpha))$-approximation algorithm for the minimum $(2,m)$-CDS, where $m\geq 2$ and $\alpha$ is the approximation ratio for the computation of a $(1,m)$-CDS. Using the solution obtained for the minimum $(1,m)$-CDS problem, they augment the connectivity of $G[C]$ by merging blocks (a block is defined as a maximal connected subgraph without a cut-vertex) of $G[C]$ recursively. When $m=2,$ this approximation ratio becomes $(4+\ln\Delta+2\ln(2+\ln\Delta)).$

In this paper, we present a different two-phase approach to the $(2,2)$-CDS problem. The first phase ends up obtaining a $C$ such that it is a 2-fold dominating set, and all connected components of $G[C]$ are biconnected ($2$-connected). The second phase, at each iteration, needs two nodes from $V\backslash C$ to reduce the number of these biconnected components by at least one. This results in an algorithm with an asymptotic approximation factor of $(3+\ln(\Delta+2)).$ By a simple modification of the potential function, our approach provides a $(3+\ln(\Delta+m))$-approximation algorithm for computing a $(2,m)$-CDS.

For related and earlier work, the reader may refer to the papers by [6] and [7].

2 Main results

Let $G=(V,E)$ be a biconnected graph. For a $C\subset V,$ define $p(C)$ to be the number of (connected) components of $G[C],$ the subgraph induced by $C.$ Define $G\langle C\rangle$ to be the spanning subgraph of $G,$ with vertex set $V,$ and edge set $\{e\in E:e$ has at least one end in $C\}.$ Let $q(C)$ represents the number of components of $G\langle C\rangle.$ For each node $v\in V$, define $m_{C}(v)$ as:

[TABLE]

Let $m(C)=\sum_{v\in V}m_{C}(v).$ Thus $m(C)$ represents the number of nodes in $V\backslash C$ which have at most one neighbor in $C.$ Note that for $m=2$, $q(C)$ and $m(C)$ are defined exactly as in [7]. Again, as in [7], we assign a color to each node in $V$ relative to a given $C$ as follows. All nodes in $C$ are colored black, nodes in $V\backslash C$ which have at least two neighbors in $C$ are colored gray, nodes in $V\backslash C$ that have exactly one neighbor in $C$ are colored red, and all other nodes are colored white.

Given $C$, we define $\hat{p}(C)$ to be

[TABLE]

in an attempt to capture the bi-connectivity deficit of $G[C]$.

A node $x_{C}$ for which the maximum in (1) is attained is called a critical node of $C.$

Finally, we use the functions, $\hat{p}(C),q(C)$ and $m(C)$ to define a potential function, $f(C)$, on $C$ as:

[TABLE]

and the difference function $\Delta_{y}f(C)$ by

[TABLE]

where $y\in V.$ We can also, equivalently, write

[TABLE]

Result. Function $f(C)$ is monotonically non-increasing. That is, $\Delta_{y}f(C)\geq 0$ for every $v$ in $V$. We need to consider three cases:

Proof: Several cases arise.

Suppose $y$ is gray. This means that $\hat{p}(C\cup\{y\})\leq\hat{p}(C)$. Clearly $m(C\cup\{y\})\leq m(C),$ and $q(C\cup\{y\})\leq q(C).$ Thus, $f(C\cup\{y\})\leq f(C),$ and hence $\Delta_{y}f(C)\geq 0.$ 2. 2.

Suppose $y$ is red. It is then connected to only one node in $C$. As $y$ is added to $C,$ its $m$-value goes down by one and its $q$-value cannot increase. Its $\hat{p}$ value may increase by $1$. Thus, $f(C\cup\{y\})\leq f(C)$. 3. 3.

Suppose $y$ is white. As $y$ is added to $C$, its $m$-value goes down by one, $q$-value goes down by at least one, and $\hat{p}$-value goes up by one. Hence, $f(C\cup\{y\})\leq f(C)$.

The following characterization of the structure of a biconnected graph [1] is useful for us.

Definition 2.1

Given a graph $H,$ we call a path $P$ an $H$-$path$ if $P$ meets $H$ exactly in its ends.

For example consider a biconnected graph that is a cycle $H$ of three nodes: $x_{1},x_{2},$ and $x_{3}.$ Then a path $P$ of three nodes $x_{1},x_{4},$ and $x_{3}$ is an $H$-path of $H.$ Adding this $H$-path to cycle $H$, keeps it biconnected. The following proposition formalizes this observation and is illustrated in Fig. 1

Proposition 2.2

[1]** A graph $H$ is biconnected if and only if it can be constructed from a cycle by successively adding $H$-$paths$ to graphs $H$ already constructed.

Suppose $C^{\ast}$ is a minimum $(2,2)$-CDS of $G.$ Since it is biconnected, using the above proposition we can list the nodes in an order such that each sublist starting from the beginning is essentially a “path”, where the first node of this “path” might correspond to a biconnected subgraph of $C^{\ast}.$ Let us illustrate this with the following example 2 of $G[C^{\ast}].$

We can list 8 nodes of this graph as the following list with sublists: $((1,2,3,4),5,6,7),8).$ Node $2$ is adjacent to node $1,$ node $3$ is adjacent to only node $1.$ Node $4,$ however, is adjacent to both nodes $2$ and node $3.$ So $(1,2,3,4)$ corresponds to a biconnected graph (cycle), and is now designated as a “single meta-node” in our list. Next, the $H$-path $(4,5,6,7,2)$ is added to this subgraph, resulting in another biconnected subgraph. Finally, adding the $H$-path $(7,8,4)$ results in $G[C^{\ast}].$ The next lemma exploits this interpretation of a biconnected graph as a “path”.

Lemma 2.3

For any two subsets $A,B\subseteq V$ and any node $y\in V$, if $B$ is a“path” then

[TABLE]

Proof:

The result is obvious if $y\in A.$ Suppose $y\in B\backslash A.$ Then the above result follows as $\Delta_{y}f(A)\geq 0$ for all $y\in V.$ Thus we assume from here on that $y\notin A\cup B.$ Define $\mu(f)=\Delta_{y}f(A\cup B)-\Delta_{y}f(A).$ It is useful to write $\mu(f)$ as:

[TABLE]

We first look at $\mu(m).$ Define $S$ to be set of nodes which are neighbors of $y$ which are white with respect to $A$ and red with respect to $B$. Let $|S|=s.$ We first want to show that:

[TABLE]

It is easy to formalize and establish this result by looking at the following two example figures: case (i): $\mu(m)=s-1,$ case (ii): $\mu(m)=s$.

In both 3(a) and 3(b) of Figure 3, $|S|=s=1.$ In Fig. 3(a), $m(A)=4,$ $m(A\cup\{y\})=3.$ Hence $\Delta_{y}m(A)=1.$ $m(A\cup B)=2,$ $m(A\cup B\cup\{y\})=1.$ Hence $\Delta_{y}m(A\cup B)=1.$ So $\mu(m)=\Delta_{y}m(A\cup B)-\Delta_{y}m(A)=0.$

In Fig. 3(b), $m(A)=4,$ $m(A\cup\{y\})=3.$ Hence $\Delta_{y}m(A)=1.$ $m(A\cup B)=3,m(A\cup B\cup\{y\})=1,$ $\Delta_{y}m(A\cup B)=2.$ Hence $\mu(m)=\Delta_{y}m(A\cup B)-\Delta_{y}m(A)=1.$

We now look at $\mu(q).$ We want to show that:

[TABLE]

Let $N_{A}(y)$ be the set of components of $G\langle A\rangle$ that are adjacent with node $y$ in $G$ (the component of $G\langle A\rangle$ containing node $y$, if any, is not counted). Then $\Delta_{y}q(A)=|N_{A}(y)|.$ Hence $\mu(q)=|N_{A\cup B}(y)|-|N_{A}(y)|.$ Again, it is easy to formalize and establish the above result by looking at the above two example figures, in Figure 3, covering the cases: 3(a), $\mu(q)\leq-s,$ and 3(b): $\mu(q)\leq-(s-1)$.

In Figure 3(a), $\Delta_{y}q(A)=|N_{A}(y)|=3,$ $\Delta_{y}q(A\cup B)=N_{A\cup B}(y)=1.$ Hence $\mu(q)=1-3=-2.$

In Figure 3(b), $\Delta_{y}q(A)=N_{A}(y)=2,$ $\Delta_{y}q(A\cup B)=|N_{A\cup B}(y)|=1.$ Hence $\mu(q)=-1.$ Zhou et al. [7] have established the above two results in a more general setting.

Now we focus on $\mu(\hat{p})=\Delta_{y}\hat{p}(A\cup B)-\Delta_{y}\hat{p}(A)$.

Let $r$ be a critical node of $G[A]$. Let $A_{r}$ be the set of nodes in the component of $G[A]$ containing node $r.$ [Note that if $G[A]$ is connected then $A_{r}=A.$]

We define three constants $\alpha$, $\beta$, and $\gamma$ in $G[A]$ as follows. Let

$\alpha$ = $p(A_{r}\backslash\{r\}),$ the number of components in $G[A_{r}\backslash\{r\}].$

$\beta$ = The number of components $G[A\backslash A_{r}]$ which are adjacent to node $y$ in $G[A]$.

$\gamma$ = The number of components of $G[A_{r}\backslash\{r\}]$ which are are adjacent to node $y$ in $G[A].~{}(\gamma\leq\alpha)$.

Refer to Fig. 4 for an illustration.

Result-1: $\Delta_{y}\hat{p}(A)=\min\{\alpha,\beta+\gamma\}-1$.

Proof: Referring to the figure above, note that $\hat{p}(A)=\alpha+\beta$. Now let us calculate $\hat{p}(A\cup\{y\})$. Whichever of the two nodes, node $r$ or node $y,$ whose removal results in the higher number of components in $G[A\cup\{y\}]$ is the critical node . Now if we remove node $r$, the resulting number of components will be $(\alpha-\gamma)+1$. If we remove node $y$ then this number is $\beta+1$. Hence

[TABLE]

Hence,

[TABLE]

Returning to $\mu(\hat{p})=\Delta_{y}\hat{p}(A\cup B)-\Delta_{y}\hat{p}(A)$, we use result-1 to make some assertions about $\mu(\hat{p})$. As we mentioned earlier, we can assume that $B$ is a set of nodes which form a “path”. Since $B$ is a “path”, adding $B$ to $A$ does not create a new critical node in $G[A\cup B]$.

Result-2: Suppose $y$ is not adjacent to $B$, then $\mu(\hat{p})=0$.

Proof: Since $y$ is not adjacent to $B$, adding $B$ to $A$ does not change $\beta$ and $\gamma$ values. $\alpha$ value may increase. Hence, $\min\{\beta+\gamma,\alpha\}$ does not change, implying $\mu(\hat{p})=0$.

Result-3: If $y$ is adjacent to $B$, then $\mu(\hat{p})\leq 1$.

Proof: If $B$ is not adjacent to $r$, then $\beta$ goes up by $1$, $\alpha$ and $\gamma$ do not change. Hence $\mu\leq 1$. If $B$ is adjacent to $r$, then both $\alpha$ and $\gamma$ go up by $1$, but $\beta$ does not change. Hence $\mu(\hat{p})=1$. Hence we have the third  inequality:

[TABLE]

Combining the three inequalities (4), (5), and (6), proves our Lemma 2.3.

Lemma 2.4

Let $G=(V,E)$ be a biconnected graph. Then, $C$ is a $2$-fold dominating set if $\Delta_{y}f(C)=0$ for every $y\in V$.

Proof: The following claims establish the proof.

Claim-1. $C\neq\emptyset$.

Suppose $C=\emptyset$. We have $\hat{p}(\emptyset)=0,~{}q(\emptyset)=|V|,~{}m(\emptyset)=|V|.$ Since $G$ is biconnected, every node in $G$ has degree at least 2. Pick any node $y.$ So $C=\{y\},$ and $\hat{p}(\{y\})=0,$ $q(\{y\})=|V|-|N_{G}(y)|,$ $m(\{y\})=|V|.$ Hence $\Delta_{y}f(C)>0,$ a contradiction.

Claim-2. $|C|\geq 3.$ Its proof is straightforward.

Claim-3. $m(C)=0.$ This claim would imply that $C$ is a 2-fold CDS.

Suppose $m(C)>0.$ This means that there is at least one node $y$ which is red or white with respect to $C.$ Suppose that $y$ is a white node. This means that $y$ is an isolated node in $G\langle C\rangle,$ and hence accounts for one component in computing $q(C).$ Adding $y$ to $C$ implies $\Delta_{y}q(C)\geq 1,$ and $\Delta_{y}m(C)=1.$ Since $\Delta_{y}\hat{p}(C)\geq-1,$ we have $\Delta_{y}f(C)\geq 1,$ a contradiction. So assume that there are no white nodes. Suppose $y$ is red. This mean that $y$ is adjacent to only one node in $C.$ Since $G$ is biconnected, $y$ is adjacent to another node $y_{1}\notin C.$ So $y_{1}$ is either red or gray. Suppose $y_{1}$ is red. Adding $y$ to $C$ makes $y_{1}$ gray. Hence $\Delta_{y}m(C)=2.$ Since $\Delta_{y}q(C)\geq 0,$ and $\Delta_{y}\hat{p}(C)\geq-1,$ we have $\Delta_{y}f(C)\geq 1,$ a contradiction. So assume $y_{1}$ is gray. Then $\Delta_{y_{1}}\hat{p}(C)\geq 0,$ $\Delta_{y_{1}}m(C)=1,$ since $\ y$ is now gray in $G[C\cup\{y_{1}\}]$, $\Delta_{y_{1}}q(C)\geq 0,$ implying $\Delta_{y_{1}}f(C)>0,$ a contradiction. This proves the claim.

Claim-4. Every (connected) component in $G[C]$ is biconnected.

To prove this, suppose $C_{1}$ is a component of $C$ which is not biconnected. Hence $C_{1}$ has a critical vertex $x$ such that $\hat{p}(C_{1})=p(C_{1}\setminus\{x\})=t\geq 2$. Since $G$ is biconnected, there exists a gray node $y$ that is connected to two different components of $G[C_{1}\setminus\{x\}]$. Hence $\hat{p}(C_{1}\cup\{y\})\leq t-1$, implying $\Delta_{y}\hat{p}(C)\geq 1$, $\Delta_{y}q(C)\geq 0$, and hence $\Delta_{y}f(C)>0$, a contradiction.

When $\Delta_{y}f(C)=0$, $\forall$ $y\in V\setminus C$, we say that phase-I of the algorithm has ended. A formal description of the Phase I algorithm is given below. At the end of Phase I, $G[C]$ has $t$ biconnected components, $t\geq 1.$ If $t=1,$ there is nothing more to do. Again, since $G$ is biconnected, if $C_{1}$ and $C_{2}$ are any two components of $G[C]$, there must exist at least two nodes $y_{1}$ and $y_{2}$ in $V\setminus C$ such that both $y_{1}$ and $y_{2}$ are connected to both $C_{1}$ and $C_{2}$, making $G[C\cup\{y_{1},y_{2}\}]$ having one less component than $G[C]$.

So, if at the end of phase-I, we have $t$ components in $G[C]$, we need to add at most $2t$ nodes to $C$ to obtain a $(2,2)$-CDS.

Theorem 2.5

The greedy algorithm with potential function $f$ for $(2,2)$-CDS is bounded by the approximation ratio $(3+\ln(\Delta+2))$, where $\Delta$ is the maximum degree of $G$.

Proof: Assume $|V|=n.$ Let $C_{G}=\{x_{1},x_{2},\dots.x_{g}\}$, in the order of nodes selected by the algorithm (phase-I). For $0\leq i\leq g$, let $C_{i}=\{x_{1},\dots.,x_{i}\}$. In particular, $C_{g}$ is the output of the algorithm. Suppose $C^{\ast}$ is a minimum $(2,2)$-CDS with $\theta=|C^{\ast}|$. Since $G[C^{\ast}]$ is biconnected, we can arrange the elements $C^{\ast}$ as $y_{1},\dots.,y_{\theta}$ such that for each $j\geq 2$, $C_{j-1}^{\ast}=\{y_{1},\dots.,y_{j-1}\}$ can be written as a “path”, such that $y_{j}$ is connected to $y_{j-1}$, and perhaps to the first node (or meta-node) of this path. If $y_{j}$ is also connected to the first “node”, then $G[{y_{1},\dots.,y_{j}}]$ is biconnected, and considered as a single meta-node. Let $C_{0}=C_{0}^{\ast}=\emptyset$. Since $f(C^{\ast})=2$, we have

[TABLE]

By the pigeonhole principle, there exists a node $y_{j}$ in $C^{\ast}$ $such$ that

[TABLE]

Since phase-I follows greedy strategy,

[TABLE]

or

[TABLE]

Denote $a_{i}=f(C_{i})-2$. Then, we can equivalently write

[TABLE]

Since all $a_{i}$’s are integers, we have

[TABLE]

Now $a_{i}>\theta$ implies $\left\lceil\frac{a_{i-1}}{\theta}\right\rceil\geq 2,$ which means $a_{i}<a_{i-1}$. So long as $a_{i}>2+\theta$, phase-I continues. Now,we can write inequality 7 as:

[TABLE]

[TABLE]

So after $\theta\ln(a_{0}/\theta)$ iteration, as in [4], $a_{i}<2\theta$. Since phase-I continues as long as $a_{i}>\theta$, after at most $\theta$ iterations $a_{i}\leq\theta$ since each iteration of phase-I reduces $a_{i}^{\prime}$ by at least one unit. Suppose phase-I ends at this stage. At this stage $f(C_{i})\leq\theta+2.$ Thus $C$ has at most $\theta+2$ biconnected components, and needs at most $2\theta+4$ additional nodes in $C$ to obtain a $(2,2)$-CDS, resulting in a bound of

[TABLE]

Asymptotically, $4/\theta$ can be ignored. So the asymptotic approximation factor is $3+\ln(\frac{a_{0}}{\theta})=3+\ln(\frac{2n}{\theta})$.  To bound $\frac{2n}{\theta},$ we proceed as follows.

Taking $i=1$, $C_{0}=\emptyset$. Then $\hat{p}(\emptyset)=0$, $q(\emptyset)=n$, $m(\emptyset)=n$. So $f(\emptyset)=2n$. $f(x_{1})=\hat{p}(\{x_{1}\})+q(\{x_{1}\})+m(\{x_{1}\})$, $\hat{p}(\{x_{1}\})=0$, $q(\{x_{1}\})=n-|N_{G}(x_{1})|-1$, $m(\{x_{1}\})=n-1$. This implies that $f(x_{1})=2n-2-|N_{G}(x_{1})|=2n-2-\Delta$. Hence

[TABLE]

Now

[TABLE]

So the approximation ratio asymptotically becomes $3+\ln(\Delta+2)$.

3 Conclusion

In this paper, we proposed a $(3+\ln(\Delta+2))$-approximation algorithm for the $(2,2)$-connected dominating set for a general graph. This algorithm can easily be generalized for the $(2,m)$-CDS problem, for $m\geq 2$, resulting in a $(3+\ln(\Delta+m))$-approximation algorithm.