Approximation algorithms and ratios for multiple domination in graphs

TL;DR

This paper analyzes approximation algorithms for classical and multiple domination numbers in graphs, providing simplified proofs and improved approximation ratios for these problems.

Contribution

It offers a shorter, self-contained proof for approximation ratios of domination problems and improves known ratios for the k-domination problem.

Findings

Approximation ratio of ln(Δ+1)+1 for classical domination.

Fixed gap in proof for k-tuple domination ratio.

Improved approximation ratio for k-domination problem.

Abstract

We analyse approximation algorithms (greedy heuristics) for the classical domination number and two multiple domination numbers in simple graphs. First, we present a short self-contained proof of the known result that the minimum domination problem in any graph $G$ with maximum degree $\Delta$ can be solved within the approximation ratio of ${\ln(\Delta+1)+1}$. The proof is based on an analysis of a simple greedy heuristic. Then, by analysing more advanced greedy heuristic techniques and using ideas from our self-contained proof for the classical domination number, we fix a gap in the existing proof of a similar result for the $k$-tuple domination number. That is, we prove that the minimum $k$-tuple domination problem indeed can be approximated within the ratio of $\ln(\Delta+1)+1$. The proof of this result is self-contained, direct, and much shorter than the existing proof, which contains the gap. Finally, we show that the known approximation ratio of $\ln(2\Delta)+1$ for the minimum $k$-domination problem can be improved to a better ratio.