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

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, 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 $α$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $α= 1$ by Ding et al. For any constant $α> 1$, we give an $O(n^{1-\frac{1}α}(\log n)^{\frac{1}α})$-approximation algorithm. When $α\geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $α=2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $ε> 0$, the problem admits no polynomial-time $2^{\log^{1-ε}n}$-approximation algorithm, improving upon the $Ω(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption).