A Polynomial Time Approximation Scheme for Euclidean Minimum Cost k-Connectivity

We present polynomial-time approximation schemes for the problem of finding a minimum-cost k-connected Euclidean graph on a finite point set in R. The cost of an edge in such a graph is equal to the Euclidean distance between its endpoints. Our schemes use Steiner points. For every given c > 1 and a set S of n points in R, a randomized version of our scheme finds an Euclidean graph on a superset of S which is k-vertex (or k-edge) connected with respect to S, and whose cost is with probability 2 within (1 + 1 c ) of the minimum cost of a k-vertex (or k-edge) connected Euclidean graph on S, in time n · (log n)(O(c √ dk))d−1 · 2((O(c √ dk)))!. We can derandomize the scheme by increasing the running time by a factor O(n). We also observe that the time cost of the derandomization of the PTA schemes for Euclidean optimization problems in R derived by Arora can be decreased by a multiplicative factor of Ω(nd−1).