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 + -) of the minimum cost of a k-vertex (or k-edge) connected Euclidean graph on 5, in time n · (log n)(O(c√dk))d-1 · 2 ((O(c√dk))d-1). 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 d derived by Arora can be decreased by a multiplicative factor of Ω(nd-1).