Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Given a Euclidean graph G in Rd with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm+n2 log n) time, resulting in a trivial O(n3m+n4 log n) time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n4) time using O(n2) space. To reduce the running time we consider several approximation algorithms.