Minimizing the Diameter of a Network Using Shortcut Edges

Abstract

We study the problem of minimizing the diameter of a graph by adding k shortcut edges, for speeding up communication in an existing network design. We develop constant-factor approximation algorithms for different variations of this problem. We also show how to improve the approximation ratios using resource augmentation to allow more than k shortcut edges. We observe a close relation between the single-source version of the problem, where we want to minimize the largest distance from a given source vertex, and the well-known k-median problem. First we show that our constant-factor approximation algorithms for the general case solve the single-source problem within a constant factor. Then, using a linear-programming formulation for the single-source version, we find a (1 + ε)[1 plus epsilon] -approximation using O(klogn) shortcut edges. To show the tightness of our result, we prove that any (3/2-e)[3/2 minus epsilon]-approximation for the single-source version must use Ω(klogn)[omega (klogn)] shortcut edges assuming P ≠ NP.