Improved Formulations and Branch-and-cut Algorithms for the Angular Constrained Minimum Spanning Tree Problem

TL;DR

This paper introduces improved integer programming formulations and branch-and-cut algorithms for the Angular Constrained Minimum Spanning Tree Problem, achieving better computational performance and new optimality certificates.

Contribution

It proposes two new formulations and associated algorithms that outperform existing methods in solving the $ ext{α}$-MSTP, with enhanced bounds and problem-solving capabilities.

Findings

BCFX$^{++}$ outperforms existing algorithms in solving $ ext{α}$-MSTP

The new formulation provides sharper bounds and more optimal solutions

Eight new optimality certificates were obtained for literature instances

Abstract

The Angular Constrained Minimum Spanning Tree Problem ($\alpha$-MSTP) is defined in terms of a complete undirected graph $G=(V,E)$ and an angle $\alpha \in (0,2\pi]$. Vertices of $G$ define points in the Euclidean plane while edges, the line segments connecting them, are weighted by the Euclidean distance between their endpoints. A spanning tree is an $\alpha$-spanning tree ($\alpha$-ST) of $G$ if, for any $i \in V$, the smallest angle that encloses all line segments corresponding to its $i$-incident edges does not exceed $\alpha$. $\alpha$-MSTP consists in finding an $\alpha$-ST with the least weight. We introduce two $\alpha-$MSTP integer programming formulations, ${\mathcal F}_{xy}^*$ and $\mathcal{F}_x^{++}$ and their accompanying Branch-and-cut (BC) algorithms, BCFXY$^*$ and BCFX$^{++}$. Both formulations can be seen as improvements over formulations coming from the literature. The strongest of them, $\mathcal{F}_x^{++}$, was obtained by: (i) lifting an existing set of inequalities in charge of enforcing $\alpha$ angular constraints and (ii) characterizing $\alpha$-MSTP valid inequalities from the Stable Set polytope, a structure behind $\alpha-$STs, that we disclosed here. These formulations and their predecessors in the literature were compared from a polyhedral perspective. From a numerical standpoint, we observed that BCFXY$^*$ and BCFX$^{++}$ compare favorably to their competitors in the literature. In fact, thanks to the quality of the bounds provided by $\mathcal{F}_x^{++}$, BCFX$^{++}$ seems to outperform the other existing $\alpha-$MSTP algorithms. It is able to solve more instances to proven optimality and to provide sharper lower bounds, when optimality is not attested within an imposed time limit. As a by-product, BCFX$^{++}$ provided 8 new optimality certificates for instances coming from the literature.