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

TL;DR

This paper studies the complexity and approximation algorithms for the minimum connected dominating set problem with routing cost constraints, providing new algorithms and hardness results for various parameters.

Contribution

It introduces new approximation algorithms for different values of the routing cost parameter and establishes stronger hardness of approximation results for the problem.

Findings

For general graphs, an $O(n^{1-1/\alpha}(\log n)^{1/\alpha})$-approximation algorithm for any constant $\alpha > 1$.

When $\alpha \geq 5$, an $O(\sqrt{n}\log n)$-approximation algorithm.

Proves that for $\alpha=2$, the problem cannot be approximated within $2^{\log^{1-\epsilon} n}$ unless $NP \subseteq DTIME(n^{poly\log n})$.

Abstract

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 $\alpha$ 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 $\alpha \geq 5$, 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 $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption).