Computing optimal shortcuts for networks

TL;DR

This paper introduces algorithms for optimally adding a segment to a Euclidean network to minimize maximum point-to-point distances, addressing the complex continuous case and providing solutions for paths.

Contribution

It presents the first polynomial-time algorithm and approximation methods for computing optimal shortcuts in a continuous network setting.

Findings

Polynomial time algorithm for general networks

Discretization approach for approximation

Improved methods for path networks with specific shortcut types

Abstract

We study augmenting a plane Euclidean network with a segment, called a shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Problems of this type have received considerable attention recently, mostly for discrete variants of the problem. We consider a fully continuous setting, where the problem of computing distances and placing a shortcut is much harder as all points on the network, instead of only the vertices, must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model: a polynomial time algorithm and a discretization of the problem that leads to an approximation algorithm. We also improve the general method for networks that are paths, restricted to two types of shortcuts: those with a fixed orientation and simple shortcuts.