On Approximation Complexity of Edge Dominating Set Problem in Dense Graphs

We study the approximation complexity of the Minimum Edge Dominating Set problem in everywhere ǫ-dense and average ǭ-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 ǫ-dense and average ǭdense graphs by using the techniques of Karpinski and Zelikovsky, the approximation ratios of min{2, 3 1+2ǫ} and of min{2, 3 3−2 √ 1−ǭ}, respectively. On the other hand, we give new approximation lower bounds for the Minimum Edge Dominating Set problem in dense graphs. Assuming the Unique Game Conjecture, we show that it is NP-hard to approximate the Minimum Edge Dominating Set problem in everywhere ǫ-dense graphs with a ratio better than 2 1+ǫ with ǫ ≥ 1/2 and 2 2− √ 1−ǭ with ǭ ≥ 3/4 in average ǭ-dense graphs. Dept. of Computer Science, University of Bonn. Work supported by Hausdorff Doctoral Fellowship. Email: [email protected] Dept. of Computer Science, University of Bonn. Work partially supported by Hausdorff Center for Mathematics, Bonn. Email: [email protected]