On approximability of the minimum-cost k-connected spanning subgraph problem

We present the first truly pojynomial-time approximation scheme (PTAS) for the mimmum-cost k-vertex(or: kedge-) connected spanning subgraph problem for complete Euclidean graphs in Rd. Previously it was known for every positive constant E how to construct in a polynomial time a graph on a superset of the input points which is k-vertex connected with respect to the input points, and whose cost is within (1 +E) of the minimum-cost of a k-vertex connected graph spanning the input points. We subsume that result by showing for every positive constant E how to construct m a polynomial-time a k-connected subgraph spanning the input points without any Steiner points and having the cost within (1 + E) of the minimum. We also studv hardness of annroximations for the minimum-cost k-v”ertexand k-edge-connected spanning subrrrauh oroblems. The. onlv inannroximabilitv result kno&*so far for the minimumrcost k-vertexand-k-edgeconnected spanning subgraph problems states that the kedge-connectivity problem in unweighted graphs does not have a PTAS unless P = NP, even for k = 2. We present a simpler proof of this result that holds even for aranhs of bounded degree, and provide the first proof that Vfnhing a PTAS for the k-vertex-connectivitv nroblem in unweiehted graphs is NP-hard even for k = 2 aid for graphs of bocnded degree. We further show that our algorithmic results for Euclidean graphs cannot be extended to arbitrarily high dimensions. -We prove that for weighted graphs tfiere-is no PTAS for the k-vertexand the k-edne-connectivitv oroblem unless P = NP, even for Euclidealgraphs in W”g ’ and k = 2.