On the Geometry of Stable Steiner Tree Instances

TL;DR

This paper investigates the geometric structure of stable Steiner tree instances under Bilu-Linial stability, showing polynomial-time solvability for 1.562-stable Euclidean instances and linking stability to approximation algorithms.

Contribution

It establishes strong geometric properties for stable instances and proves polynomial-time solvability for 1.562-stable Euclidean Steiner trees, enhancing understanding of stability in this problem.

Findings

Stable instances satisfy specific geometric properties.

1.562-stable Euclidean Steiner trees are polynomial-time solvable.

Connection established between approximation algorithms and stability.

Abstract

In this note we consider the Steiner tree problem under Bilu-Linial stability. We give strong geometric structural properties that need to be satisfied by stable instances. We then make use of, and strengthen, these geometric properties to show that $1.562$-stable instances of Euclidean Steiner trees are polynomial-time solvable. We also provide a connection between certain approximation algorithms and Bilu-Linial stability for Steiner trees.