On the Complexity of Edge Packing and Vertex Packing

This paper studies the computational complexity of the Edge Packing problem and the Vertex Packing problem. The edge packing problem (denoted by EDS ) and the vertex packing problem (denoted by DS ) are linear programming duals of the edge dominating set problem and the dominating set problem respectively. It is shown that these two problems are equivalent to the set packing problem with respect to hardness of approximation and parametric complexity. It follows that EDS and DS cannot be approximated asymptotically within a factor of O(|V | 1 2−ǫ ) for any ǫ > 0 unless NP = ZPP where, V is the set of vertices of the given graph. This is in contrast with the fact that the edge dominating set problem is 2-approximable whereas the dominating set problem is known to have an O(log |V |) approximation algorithm. It also follows from our proof that EDS and DS are W[1]-complete under standard parameterization.