A polynomial time approximation scheme for euclidean minimum cost k-connectivity

Abstract

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 d . 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 ℝd, 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 1/2 within (1 + 1/c) of the minimum cost of a k-vertex (or k-edge) connected Euclidean graph on S, in time \(n \cdot \left( {logn} \right)^{\left( {\mathcal{O}\left( {c\sqrt d k} \right)} \right)^{d - 1} } \cdot 2^{\left( {\left( {\mathcal{O}\left( {c\sqrt d k} \right)} \right)^{d - 1} } \right)!}\). 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 ℝd derived by Arora can be decreased by a multiplicative factor of Ω(n d−1 ).