Exact and Heuristic Algorithms for the Domination Problem

TL;DR

This paper introduces new exact and heuristic algorithms for the NP-hard domination problem in graphs, demonstrating improved performance and solution quality over existing methods through computational experiments.

Contribution

It formulates the domination problem as an ILP, compares with existing algorithms, and proposes new algorithms that outperform previous methods in speed and solution quality.

Findings

Exact algorithm outperforms previous exact methods on middle-dense graphs.

Heuristic algorithm is faster and finds optimal solutions for over half of the instances.

Heuristic improves known solutions with an average approximation error of 1.18.

Abstract

In a simple connected graph $G=(V,E)$, a subset of vertices $S \subseteq V$ is a dominating set if any vertex $v \in V\setminus S$ is adjacent to some vertex $x$ from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of $1.18$.