Algorithms and Computational Complexity
Determining whether there is an edge dominating set of a given size for a given graph is an NP-complete problem (and therefore finding a minimum edge dominating set is an NP-hard problem). Yannakakis & Gavril (1980) show that the problem is NP-complete even in the case of a bipartite graph with maximum degree 3, and also in the case of a planar graph with maximum degree 3.
There is a simple polynomial-time approximation algorithm with approximation factor 2: find any maximal matching. A maximal matching is an edge dominating set; furthermore, a maximal matching M can be at worst 2 times as large as a smallest maximal matching, and a smallest maximal matching has the same size as the smallest edge dominating set.
Also the edge-weighted version of the problem can be approximated within factor 2, but the algorithm is considerably more complicated (Fujito & Nagamochi 2002).
Chlebík & Chlebíková (2006) show that finding a better than (7/6)-approximation is NP-hard.
Read more about this topic: Edge Dominating Set
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