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 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).