# Improved approximation algorithms for $k$-connected $m$-dominating set problems

## TL;DR

This paper presents improved approximation algorithms for the $k$-connected $m$-dominating set problem, achieving better ratios for unit disc-graphs and the first non-trivial ratio for general graphs, enhancing efficiency in network design.

## Contribution

The authors develop new approximation algorithms with improved ratios for the $(k,m)$-connected dominating set problem in different graph classes, advancing prior work.

## Key findings

- Achieved $O(k \\ln k)$ approximation for unit disc-graphs.
- Established $O(k^2 \\ln n)$ approximation for general graphs.
- Improved upon previous $O(k^2 \\ln k)$ ratio for unit disc-graphs.

## Abstract

A graph is $k$-connected if it has $k$ internally-disjoint paths between every pair of nodes. A subset $S$ of nodes in a graph $G$ is a $k$-connected set if the subgraph $G[S]$ induced by $S$ is $k$-connected; $S$ is an $m$-dominating set if every $v \in V \setminus S$ has at least $m$ neighbors in $S$. If $S$ is both $k$-connected and $m$-dominating then $S$ is a $k$-connected $m$-dominating set, or $(k,m)$-cds for short. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. We consider the case $m \geq k$ and obtain the following approximation ratios. For unit disc-graphs we obtain ratio $O(k\ln k)$, improving the previous ratio $O(k^2 \ln k)$. For general graphs we obtain the first non-trivial approximation ratio $O(k^2 \ln n)$.