Approximating Edge Dominating Set in Dense Graphs

We study the approximation complexity of the Minimum Edge Dominating Set problem in everywhere epsilon-dense and average (epsilon) over bar -dense graphs. More precisely, we consider the computational complexity of approximating a generalization of the Minimum Edge Dominating Set problem, the so called Minimum Subset Edge Dominating Set problem. As a direct result, we obtain for the special case of the Minimum Edge Dominating Set problem in everywhere epsilon-dense and average (epsilon) over bar -dense graphs by using the techniques of Karpinski and Zelikovsky, the approximation ratios of min{2, 3/(1+ 2 epsilon)} and of min{2, 3/(3 - 2 root 1 - (epsilon) over bar)}, respectively. On the other hand, we show that it is UGC-hard to approximate the Minimum Edge Dominating Set problem in everywhere epsilon-dense graphs with a ratio better than 2/(1 + epsilon) with epsilon > 1/3 and 2/(2 - root 1 - (epsilon) over bar) with (epsilon) over bar > 5/9 in average (epsilon) over bar -dense graphs.