Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees

TL;DR

This paper introduces efficient algorithms for adding a single edge to a path or tree to minimize the graph's diameter, including exact and approximation methods with improved running times.

Contribution

It provides the first exact algorithms for diameter optimization in paths and trees, and a fast approximation algorithm for paths in fixed-dimensional Euclidean space.

Findings

Exact algorithm for paths in O(n log^3 n) time.

Exact algorithm for trees in O(n^2 log n) time.

Approximation algorithm with (1+ε) factor in O(n + 1/ε^3) time.

Abstract

We consider the problem of augmenting an n-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input graph is a path, running in O(n \log^3 n) time, and (ii) the input graph is a tree, running in O(n^2 \log n) time. We also present an algorithm that computes a (1+\eps)-approximation in O(n + 1/\eps^3) time, for paths in R^d, where d is a constant.