Fixed Topology Minimum-Length Trees with Neighborhoods

TL;DR

This paper introduces a new optimization problem for embedding tree-shaped graphs into metric spaces with restricted areas, aiming to minimize total length, with applications in engineering routing.

Contribution

It presents novel mathematical optimization formulations and a data-driven approach for solving the Fixed Topology Minimum-Length Tree with Neighborhood problem.

Findings

Effective optimization formulations for continuous and network embeddings.

Dimensionality reduction improves computational efficiency.

Validated through extensive computational experiments.

Abstract

In this paper, we introduce the Fixed Topology Minimum-Length Tree with Neighborhood Problem, which aims to embed a rooted tree-shaped graph into a $d$-dimensional metric space while minimizing its total length provided that the nodes must be embedded to some restricted areas. This problem has significant applications in efficiently routing cables or pipelines in engineering designs. We propose novel mathematical optimization-based approaches to solve different versions of the problem based on the domain for the embedding. In cases where the embedding maps to a continuous space, we provide several Mixed Integer Nonlinear Optimization formulations. If the embedding is to a network, we derive a mixed integer linear programming formulation as well as a dimensionality reduction methodology that allows for solving larger problems in less CPU time. A data-driven methodology is also proposed to construct a proper network based on the instance of the problem. We report the results of a battery of computational experiments that validate our proposal.