Abstract
Given an undirected graph on a node set V and positive integers k and m, a kconnected mdominating 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 kconnected subgraph is induced by S. The weighted (k, m)CDS problem is to find a minimum weight (k, m)CDS in a given nodeweighted 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 primaldual \(O(k^2 \log k)\)approximation algorithm for the weighted (k, m)CDS problem.
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Acknowledgements
The author thanks anonymous referees for their careful reading and many useful comments. This work was supported by JST ERATO Grant No. JPMJER1201, and JSPS KAKENHI Grant No. 17K00040.
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Appendix: Improved Analysis of Shang et al. for the Unweighted (2, m)CDS
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 \(135/m\) 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 \(G[V \setminus \bigcup _{i'=0}^{i1} I_{i'}]\) 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 idominating 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 cutnode of G[T] is included in \(I_1\) or in C.

(iv)
\(T \le (5+5/m)\mathrm {OPT}\) for \(m\le 5\), and \(T \le (75/m)\mathrm {OPT}\) 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 2connected as follows. First, S is initialized to be an empty set. Then, the algorithm computes a Tcut X such that \(\varGamma _{T}(X)=1\) and no Tpath in \(G[T \cup S]\) covers X. For this Tcut X, the algorithm finds a Tpath 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 2connected, 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 cutnodes in G[T], which is at most \(I_1 + C \le 2I_1 \le 2\max \{5/m,1\}\mathrm {OPT}\), 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 cutnodes in G[T]. This can be observed by considering a star with n leaves. The star has only one cutnode. To make it 2connected, we need to add \(n1\) paths to connect the leaves.
We claim that a correct upperbound 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 cutnodes. Each block consists of at least two nodes. If a block consists of more than nodes, then it is a 2connected 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 cutnode 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 cutnode 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 cutnode w.
In each iteration of the algorithm, a Tpath 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 Tpath 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 cutnode of G[T], and let b and \(b'\) be blocks that include x. When the algorithm chooses a Tpath 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 2connected if virtual edges induce a connected graph on the set of neighbors of each cutnode x in F.
Proof
Suppose that \(T\cup S\) is not 2connected even if the condition holds. Then there exists a cutnode 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 mdominating and \(m \ge 2\), we can assume that \(x,y,y' \in T\). Hence x is also a cutnode 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 cutnode 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 cutnode 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 cutnode 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 cutnode 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 \(I_1 + C1 \le 2\max \{5/m,1\}\mathrm {OPT}1\), 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 \(135/m\) for \(m\ge 6\).
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Fukunaga, T. Approximation Algorithms for Highly Connected Multidominating Sets in Unit Disk Graphs. Algorithmica 80, 3270–3292 (2018). https://doi.org/10.1007/s0045301703852
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Keywords
 Connected dominating set
 Unit disk graph
 Approximation algorithm