Exact and Approximation Algorithms for the Domination Problem

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 including facility location problems can be modeled using this problem, known to be among the difficult NP-hard problems in its class. We propose exact enumeration and approximation algorithms for the domination problem. The exact algorithm has solved optimally problem instances with over 1000 vertices within 6 minutes. This is a drastic breakthrough compared to the earlier known exact state-of-the-art algorithm which was capable to solve the instances up to 300 vertices within the range of 8 hours. Among the instances that were solved by both algorithms, in average, our exact algorithm was about 170 times faster than the former state-of-the-art algorithm. Our approximation algorithm, in 98.62% of the analyzed instances, improved the earlier known state-of-the-art solutions. It was able to solve problem instances with more than 2000 vertices in less than 1 minute, whereas it found an optimal solution for 61.54% of the instances. For the instances where the optimum was not found, the approximation error was $1.18$.