On Approximability of the Minimum-Cost k-Connected Spanning Subgraph Problem

We present the rst truly polynomial-time approximation scheme (PTAS) for the minimum-cost k-vertex-(or, k-edge-) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " 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+") of the minimum-cost of a k-vertex connected graph spanning the input points. We subsume that result by showing for every positive constant " how to construct in a polynomial-time a k-connected subgraph spanning the input points without any Steiner points and having the cost within (1 + ") of the minimum. We also study hardness of approximations for the minimum-cost k-vertex-and k-edge-connected spanning subgraph problems. The only inapproximability result known so far for the minimum-cost k-vertex-and k-edge-connected spanning subgraph problems states that the k-edge-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 graphs of bounded degree, and provide the rst proof that nding a PTAS for the k-vertex-connectivity problem in unweighted graphs is NP-hard even for k = 2 and for graphs of bounded degree. We further show that our algorithmic results for Euclidean graphs cannot be extended to arbitrarily high dimensions. We prove that for weighted graphs there is no PTAS for the k-vertex-and the k-edge-connectivity problem unless P = NP; even for Euclidean graphs in R log n and k = 2: