A $${{10}}$$ -Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem

AbstractWe study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edge-dominating set to edge cover. Our main result is a simple $$2\frac{1}{{10}}$$ -approximation algorithm for the weighted edge-dominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2rWVC, where rWVC is the approximation guarantee of any polynomial-time weighted vertex cover algorithm. The best value of rWVC currently stands at $$2 - \frac{{\log \log \left| V \right|}}{{2\log \left| V \right|}}$$ . Furthermore we establish that the factor of $$2\frac{1}{{10}}$$ is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of the problem.