A New Approximation Algorithm for Minimum-Weight $(1,m)$--Connected Dominating Set

TL;DR

This paper introduces a novel polynomial-time approximation algorithm for the minimum-weight (1,m)-connected dominating set problem in graphs, achieving a ratio based on maximum degree and harmonic numbers, improving solution efficiency.

Contribution

The paper presents a new approximation algorithm with a specific ratio for the minimum-weight (1,m)-connected dominating set problem, advancing existing methods.

Findings

Approximation ratio of 2H(δ_max + m - 1) achieved

Algorithm runs in polynomial time

Improves bounds for minimum-weight (1,m)-CDS problem

Abstract

Consider a graph with nonnegative node weight. A vertex subset is called a CDS (connected dominating set) if every other node has at least one neighbor in the subset and the subset induces a connected subgraph. Furthermore, if every other node has at least $m$ neighbors in the subset, then the node subset is called a $(1,m)$CDS. The minimum-weight $(1,m)$CDS problem aims at finding a $(1,m)$CDS with minimum total node weight. In this paper, we present a new polynomial-time approximation algorithm for this problem with approximation ratio $2H(\delta_{\max}+m-1)$, where $\delta_{\max}$ is the maximum degree of the given graph and $H(\cdot)$ is the Harmonic function, i.e., $H(k)=\sum_{i=1}^k \frac{1}{i}$.