On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

Abstract

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$ and a positive integer $\alpha$, and the goal is to find the smallest connected dominating set D of G such that, for any two non-adjacent vertices u and v in G, the number of internal nodes on the shortest path between u and v in the subgraph of G induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in G. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha }}(\log n)^{\frac{1}{\alpha }})$-approximation algorithm. When $\alpha \ge 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log ^{1-\epsilon }n}$-approximation algorithm, improving upon the $\varOmega (\log \delta )$ bound by Du et al., where $\delta$ is the maximum degree of G (albeit under a stronger hardness assumption).

This work is partially supported by the Ministry of Science and Technology of R.O.C. under contract No. MOST 106-2221-E-004-005-MY3.

Appendices

A Proof of Lemma 6

If [u, v] is in $[PX, PX \cup \{h_{X,R}, h_{Y,R}\}]$, $[PY, PY \cup \{h_{X,R}, h_{Y,R}\}]$, $[X, X \cup R \cup \{h_{PX}, h_{PY}\}]$, $[Y, Y \cup R \cup \{h_{PX}, h_{PY}\}]$, or $[R, R \cup \{h_{PX}, h_{PY}\}]$, then [u, v] can be covered by one vertex in H. If [u, v] is in $[PX,Y], [PY,X], [X, \{h_{Y,R}\}]$, or $[Y,\{h_{X,R}\}]$, then [u, v] can be covered by an edge in H. If [u, v] is in $[PX, X \cup R \cup \{h_{PX}, h_{PY}\}]$, $[PY, Y \cup R \cup \{h_{PX}, h_{PY}\}]$, or [X, Y], then [u, v] cannot be a target couple (since u and v do not have a common neighbor). If [u, v] is in $[X,\{h_{X,R}\}], [Y,\{h_{Y,R}\}]$, or $[R, \{h_{X,R}, h_{Y,R}\}]$, then [u, v] cannot be a target couple (since u and v are adjacent⁶). Moreover, it is easy to see that H covers all the target couples in [H, H] or $[V(G'), M]$, where $V(G')$ is the vertex set of $G'$. Finally, observe that if [u, v] is in [PX, PY], then H cannot cover [u, v].     $\square$

B Reduction from the 1-DR-$\alpha$ Problem to Other Related Problems

In this section, we show that the 1-DR-$\alpha$ problem can be transformed into the submodular cost set cover problem and the minimum rainbow subgraph problem on multigraphs. We also summarize the approximability results of these two problems in the literature. Future progress in the approximability results of these two problems may lead to better approximation algorithms for the 1-DR-$\alpha$ problem.

Submodular Cost Set Cover Problem: The 1-DR-$\alpha$ problem can be considered as a special case of the submodular cost set cover problem [10, 14, 23]. In the set cover problem, we are given a set of targets $\mathcal {T}$ and a set of objects $\mathcal {S}$. Each object in $\mathcal {S}$ can cover a subset of $\mathcal {T}$ (specified in the input). The goal is to choose the smallest subset of $\mathcal {S}$ that covers $\mathcal {T}$. In the submodular cost set cover problem, there is a non-negative submodular function c that maps each subset of $\mathcal {S}$ to a cost, and the goal is to find the set cover with the minimum cost. To transform the 1-DR-$\alpha$ problem with input $G=(V,E)$ to the submodular cost set cover problem, let $\mathcal {T}$ be the union of V and the set of all target couples, and let $\mathcal {S}$ be the set of all subsets of V with size at most $\alpha$. Hence, each object in $\mathcal {S}$ is a subset of V. An object $S \in \mathcal {S}$ can cover a vertex v if v is adjacent to some vertex in S or $v \in S$. An object $S \in \mathcal {S}$ can cover a target couple [u, v] if $m^S(u,v) \le \alpha$. The cost of a subset $\mathcal {S}'$ of $\mathcal {S}$ is simply the size of the union of objects in $\mathcal {S}'$, i.e., the number of distinct vertices specified in $\mathcal {S}'$.

Iwata and Nagano proposed a $|\mathcal {T}|$-approximation algorithm and an f-approximation algorithm, where f is the maximum frequency, $\max _{T\in \mathcal {T}} |\{S \in \mathcal {S}| S \text { covers } T\}|$ [14]. Koufogiannakis and Young also proposed an f-approximation algorithm when the cost function c is non-decreasing [18]. It is easy to see that these algorithms give trivial bounds for the 1-DR-$\alpha$ problem. When the cost function c is integer-valued, non-decreasing, and satisfies $c(\emptyset )=0$, Wan et al. proposed a $\rho H(\gamma )$-approximation algorithm, where $\rho = \min \limits _{\mathcal {S}^*: \mathcal {S}^* \text { is an optimal solution}} {\frac{\sum _{S \in \mathcal {S}^*}c(\{S\})}{c(\mathcal {S}^*)}}$, $\gamma$ is the largest number of targets that can be covered by an object in $\mathcal {S}$, and H(k) is the k-th Harmonic number [23]. Du et al. applied this algorithm to the 1-DR-$\alpha$ problem on UDG for $\alpha \ge 5$ and obtained a constant factor approximation algorithm [10]. It is unclear whether or not $\rho$ can be upper bounded by $O(n^{1-\epsilon })$ for some $\epsilon > 0$ when applied to the 1-DR-$\alpha$ problem on general graphs.

Minimum Rainbow Subgraph Problem on Multigraphs: Given a set of p colors and a multigraph H, where each edge is colored with one of the p colors, the Minimum Rainbow Subgraph (MRS) problem asks for the smallest vertex subset D of H, such that each of the p colors appears in some edge induced by D. The 1-DR-2 problem can be transformed into the MRS problem as follows. Let $G=(V,E)$ be the input graph of the 1-DR-2 problem. Let T be the union of V and the set of all target couples. The set of colors for the MRS problem is $\{c_i|i \in T\}$. The input multigraph H of the MRS problem has the same vertex set as G. To form a dominating set, for each $v \in V$, v is incident to $d(v)+1$ loops (v, v) in H, where d(v) is the degree of v in G. Each of these loops receives a different color in $\{c_v\} \cup \{c_u| (u,v) \in E\}$. For each target couple [u, v] in G, if w is a common neighbor of u and v in G, we add a loop (w, w) with color $c_{[u,v]}$ to H. Finally, for each target couple [u, v] in G, if $(u,w_1, w_2,v)$ is a path in G, we add an edge $(w_1, w_2)$ with color $c_{[u,v]}$ to H. The MRS problem can be transformed into the SCP problem. When the input graph is simple, Tirodkar and Vishwanathan proposed an $O(n^{1/3}\log n)$-approximation algorithm [22].