A 21/10-approximation algorithm for a generalization of the weighted edge-dominating set problem

We study the approximability of the weighted edge-dominating set problem. A lthough 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 clos e relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showi ng that weighted edge domination is as hard to approximate as the well stud ied weighted vertex cover problem, we consider a natural strategy, reducing edge-dominating set to edge cover. Our main result is a simple 2 1/10-appr oximation algorithm for the weighted edge-dominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2r(WVC), where r(WVC) is the approximation guarantee of any polynomial-tim e weighted vertex cover algorithm. The best value of r(WVC) currently stand s at 2-log log |V|/2 log |V|. Furthermore we establish that the factor of 2 1/10 is tight in the sense that it coincides with the integrality gap incu rred by a natural linear programming relaxation of the problem.