A polynomial-time approximation to a minimum dominating set in a graph

TL;DR

This paper presents a polynomial-time approximation algorithm for the minimum dominating set problem in graphs, combining greedy construction and a purification process to improve solution quality efficiently.

Contribution

It introduces a two-stage approximation algorithm with analysis, including conditions for optimality and new approximation ratios, extending prior greedy methods.

Findings

The algorithm performs well on certain graph classes, often producing optimal solutions.

The second purification stage significantly reduces the dominating set size in practice.

Experimental results show a notable gap between initial and refined solutions.

Abstract

A {\em dominating set} of a graph $G=(V,E)$ is a subset of vertices $S\subseteq V$ such that every vertex $v\in V\setminus S$ has at least one neighbor in $S$. Finding a dominating set with the minimum cardinality in a connected graph $G=(V,E)$ is known to be NP-hard. A polynomial-time approximation algorithm for this problem, described here, works in two stages. At the first stage a dominant set is generated by a greedy algorithm, and at the second stage this dominating set is purified (reduced). The reduction is achieved by the analysis of the flowchart of the algorithm of the first stage and a special kind of clustering of the dominating set generated at the first stage. The clustering of the dominating set naturally leads to a special kind of a spanning forest of graph $G$, which serves as a basis for the second purification stage. We expose some types of graphs for which the algorithm of the first stage already delivers an optimal solution and derive sufficient conditions when the overall algorithm constructs an optimal solution. We give three alternative approximation ratios for the algorithm of the first stage, two of which are expressed in terms of solely invariant problem instance parameters, and we also give one additional approximation ratio for the overall two-stage algorithm. The greedy algorithm of the first stage turned out to be essentially the same as the earlier known state-of-the-art algorithms for the set cover and dominating set problem Chv\'atal \cite{chvatal} and Parekh \cite{parekh}. The second purification stage results in a significant reduction of the dominant set created at the first stage, in practice. The practical behavior of both stages was verified for randomly generated problem instances. The computational experiments emphasize the gap between a solution of Stage 1 and a solution of Stage 2.