Optimum adaptation of a Steiner network

TL;DR

This paper presents a first-order approximation method for efficiently updating Steiner tree solutions after perturbations in terminal node positions, with numerical validation and analysis of its effectiveness and limitations.

Contribution

It introduces a novel first-order approximation theorem for adapting Steiner trees to perturbed terminal positions, improving solution update efficiency.

Findings

The approximation effectively updates Steiner trees after small perturbations.

Numerical examples demonstrate the method's effectiveness and limitations.

Stepwise application handles larger perturbations successfully.

Abstract

The Euclidean Steiner tree problem, normally posed in two dimensions, seeks to connect a set of prescribed terminal nodes by placing additional nodes, known as Steiner points, with edges connecting such nodes either to another Steiner point or a terminal node, and with the placements minimising the sum of all the edge lengths of the associated tree. We consider a problem in which we start with a known solution to a Steiner tree problem, and the terminal positions are then perturbed. A first-order approximation theorem is established for efficiently updating the Steiner point positions to recover a Steiner tree solution after the perturbations to terminal nodes. Numerical examples illustrate the effectiveness of our approach (including a stepwise application for large perturbations) as well as its limitations.