Approximation Algorithms for Highly Connected Multi-dominating Sets in Unit Disk Graphs

Abstract

Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((k, m)-CDS) is defined as a subset S of V such that each node in $V{\setminus }S$ has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (k, m)-CDS problem is to find a minimum weight (k, m)-CDS in a given node-weighted graph. The problem is called the unweighted (k, m)-CDS problem if the objective is to minimize the cardinality of a (k, m)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. In this paper, we consider the case in which $k \le m$, and we present a simple $O(k5^k)$-approximation algorithm for the unweighted (k, m)-CDS problem, and a primal-dual $O(k^2 \log k)$-approximation algorithm for the weighted (k, m)-CDS problem.

Appendix: Improved Analysis of Shang et al. for the Unweighted (2, m)-CDS

Shang et al. [21] gave an approximation algorithm for the unweighted (2, m)-CDS problem. They claimed that their algorithm achieves an approximation factor of $5+25/m$ for $2 \le m \le 5$, and 11 for $m \ge 6$. However, Shi et al. [22] pointed out that its analysis contains a flaw. Shi et al. [22] also rectified the analysis, and showed that its approximation factor is $15+15/m$ for $2 \le m \le 5$ and 21 for $m \ge 6$. Simultaneously, Shi et al. [22] presented an algorithm for the unweighted (2, m)-CDS problem on general graphs, and proved that their algorithm achieves approximation factors $7+5/m+2\ln (5+5/m)$ for $2 \le m\le 5$, and 11 for $m \ge 6$ if the graphs are restricted to unit disk graphs. In this section, we present an improved analysis of Shang et al. [21]. It gives approximation factors $5+35/m$ for $2 \le m \le 5$, and $13-5/m$ for $m\ge 6$. Although these are not better than those of Shi et al. [22], we believe that it is worth noting.

Let us begin with illustrating the algorithm of Shang et al. [21]. Let $\mathrm {OPT}$ denote the minimum size of (2, m)-CDSs. Let $I_{i}$ be a maximal independent set of $G[V \setminus \bigcup _{i'=0}^{i-1} I_{i'}]$ for each $i =1,\ldots ,m$, where $I_0=\emptyset$. The algorithm first computes $I_1,\ldots ,I_m$, and $C \subseteq V{\setminus }I_1$ such that $|C|\le |I_1|$ and $G[C \cup I_1]$ is connected. The following properties are proven.

• (i) $|I_i| \le \max \{5/m,1\}\mathrm {OPT}$ for each $i=1,\ldots ,m$.

• (ii) $\bigcup _{i'=1}^{i}I_{i'}$ is an i-dominating set for each $i=1,\ldots ,m$, and hence $T:= \bigcup _{i=1}^m I_i \cup C$ is a (1, m)-CDS.

• (iii) Each cut-node of G[T] is included in $I_1$ or in C.

• (iv) $|T| \le (5+5/m)\mathrm {OPT}$ for $m\le 5$, and $|T| \le (7-5/m)\mathrm {OPT}$ for $m \ge 6$ (this is slightly better than the conclusion in [21], but they proved this).

After this step, the algorithm computes a node set S such that $T\cup S$ is 2-connected as follows. First, S is initialized to be an empty set. Then, the algorithm computes a T-cut X such that $|\varGamma _{T}(X)|=1$ and no T-path in $G[T \cup S]$ covers X. For this T-cut X, the algorithm finds a T-path that covers X with at most two inner nodes, and it adds these inner nodes to the solution S. This procedure is repeated until $T \cup S$ becomes 2-connected, and the algorithm outputs $T\cup S$ as a (2, m)-CDS.

Shang et al. [21] claimed that the number of iterations is at most the number of cut-nodes in G[T], which is at most $|I_1| + |C| \le 2|I_1| \le 2\max \{5/m,1\}\mathrm {OPT}$, due to properties (i) and (iii). Since each iteration adds at most two nodes, when the algorithm terminates, $|S| \le 4\max \{5/m,1\}\mathrm {OPT}$. This is their analysis of the algorithm.

However, the number of iterations is not bounded by the number of cut-nodes in G[T]. This can be observed by considering a star with n leaves. The star has only one cut-node. To make it 2-connected, we need to add $n-1$ paths to connect the leaves.

We claim that a correct upper-bound on the number of iterations is the number of blocks of G[T], and this number is bounded by $|I_1|+|C|+|I_2| -1 \le 3\max \{5/m,1\}\mathrm {OPT}-1$. A block of a connected graph is a maximal set of nodes that induces a connected subgraph without cut-nodes. Each block consists of at least two nodes. If a block consists of more than nodes, then it is a 2-connected component of the graph. If a block consists of only two nodes, then it induces an edge.

For proving our claim, let us consider the tree F that represents the block decomposition of G[T]. Namely, the node set of F is the disjoint union of two node sets B and W. Each node in B corresponds to a block of G[T], and each node in W corresponds to a cut-node of G[T]. In what follows, we identify a node in B with the corresponding block of G[T], and we identify a node in W with the corresponding cut-node of G[T]. A node $b \in B$ and a node $w \in W$ are joined by an edge in F when the block b includes the cut-node w.

In each iteration of the algorithm, a T-path is selected to join two different blocks of G[T], and the inner nodes of the path are added to the solution S. Let u and v denote the end nodes of a T-path P, and let $\rho _u$ be a block that includes u. If more than one block includes u, we let $\rho _u$ denote the one nearest to the blocks including v on F; $\rho _v$ is defined in the same way. Let x be a cut-node of G[T], and let b and $b'$ be blocks that include x. When the algorithm chooses a T-path P, we add a virtual edge that joins b and $b'$ if x, b, and $b'$ are on the path between $\rho _u$ and $\rho _v$ on F.

Lemma 14

$T\cup S$ is 2-connected if virtual edges induce a connected graph on the set of neighbors of each cut-node x in F.

Proof

Suppose that $T\cup S$ is not 2-connected even if the condition holds. Then there exists a cut-node x of $G[T\cup S]$. In other words, after removing x from $G[T\cup S]$, some neighbors y and $y'$ of x are included in the different connected component. Since T is m-dominating and $m \ge 2$, we can assume that $x,y,y' \in T$. Hence x is also a cut-node of G[T].

Let b and $b'$ be the blocks of G[T] including y and $y'$, respectively. Since y and $y'$ are neighbors of x, both b and $b'$ include x (i.e., b and $b'$ are neighbors of x in F). By the condition of the lemma, b and $b'$ are connected by a path of the virtual edges. This implies that there exists a path on $G[T\cup S]$ that connects y and $y'$, and that does not pass through x. Hence, x is not a cut-node in $G[T\cup S]$, which is a contradiction.$\square$

The following lemma presents a bound on the number of iterations in this algorithm.

Lemma 15

The number of iterations is at most $3\max \{5/m,1\}\mathrm {OPT}-1$.

Proof

Let x be a cut-node on F, and let $\psi _x$ denote the number of connected components induced by the virtual edges on the neighbor set of x. In each iteration of the algorithm, $\psi _x$ is decreased by at least one for some cut-node x. When the first iteration begins, $\psi _x$ is equal to the degree of x in F. Since all leaves in F are included in B, $\sum _{x \in W} \psi _x=|B|+|W|$ holds at the beginning of the first iteration. The iterations terminate when $\sum _{x \in W} \psi _x=|W|$. Hence, the number of iterations is at most |B|. We will determine |B| below.

Let H be a spanning tree on $G[I_1 \cup C]$. We show that each block contains a node in $I_2{\setminus }C$ or an edge in H. To see this, suppose that a block b of G[T] contains no edge in H. If b contains more than one node in $I_1 \cup C$, then it includes an edge in H. Since, by the assumption, this does not happen, b includes at most one node in $I_1 \cup C$. Hence, there exists a node $v \in \bigcup _{i=2}^m I_i{\setminus }C$ in b. If $v \in I_2$, we are done. Suppose that $v \not \in I_2$. By property (ii), v has neighbors in $I_1$ and in $I_2$. By property (iii), v is not a cut-node in G[T], so these neighbors must be inside b. If the neighbor in $I_2$ is included in C, b contains two nodes in $I_1 \cup C$. Since this contradicts the assumption, b contains a node in $I_2{\setminus }C$.

Nodes in $\bigcup _{i=2}^m I_2{\setminus }C$ and edges in G[T] are not contained in more than one block of G[T]. The number of edges in H is $|I_1| + |C|-1 \le 2\max \{5/m,1\}\mathrm {OPT}-1$, and $|I_2| \le \max \{5/m,1\}\mathrm {OPT}$ by property (i). Hence $|B| \le 3\max \{5/m,1\}\mathrm {OPT}-1$.$\square$

From Lemma 15 and property (iv), we obtain the following approximation guarantee for the algorithm.

Theorem 4

The algorithm of Shang et al. [21] attains an approximation factor $5+35/m$ for $2 \le m \le 5$, and $13-5/m$ for $m\ge 6$.