Reorganizing topologies of Steiner trees to accelerate their elimination

TL;DR

This paper introduces a novel topology reorganization technique for Steiner trees that accelerates their elimination in optimization algorithms, improves exploration strategies, and corrects previous lower bound usage, with potential applications in higher dimensions.

Contribution

It presents a new method for reorganizing Steiner tree topologies within enumeration schemes, enhancing the efficiency of exact algorithms and addressing prior lower bound inaccuracies.

Findings

Topology reorganizations can improve branch-and-bound exploration.

Corrected the use of pre-optimization lower bounds.

Demonstrated potential of twin trees in higher dimensions.

Abstract

We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in $d$-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.