# Approximating $k$-connected $m$-dominating sets

## TL;DR

This paper develops improved approximation algorithms for the $k$-connected $m$-dominating set problem in graphs, achieving better ratios especially for unit disc graphs and uniform weights, advancing the theoretical understanding of this problem.

## Contribution

It introduces new approximation ratios for the $(k,m)$-CDS problem, notably reducing the ratio for general graphs to $O(k
ln n)$ and providing the first sublinear and polylogarithmic ratios for unit disc graphs.

## Key findings

- Improved approximation ratio $O(k
ln n)$ for general graphs.
- First sublinear ratio for the problem in unit disc graphs.
- Polylogarithmic ratio $O(
ln^2 k)/
ln$ for $m extgreater= (1+
ln)k$.

## Abstract

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. 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. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln k)$ to $\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k)$ -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln^2 k)/\epsilon$ when $m \geq (1+\epsilon)k$; furthermore, we obtain ratio $\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k)$ for uniform weights. These results are obtained by showing the same ratios for the Subset $k$-Connectivity problem when the set $T$ of terminals is an $m$-dominating set with $m \geq k$.