Improved Formulations and Branch-and-cut Algorithms for the Angular Constrained Minimum Spanning Tree Problem
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The Angular Constrained Minimum Spanning Tree Problem (α-MSTP) is defined in terms of a complete undirected graph G=(V,E) and an angle α∈ (0,2π]. 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 α-spanning tree (α-ST) of G if, for any i ∈ V, the smallest angle that encloses all line segments corresponding to its i-incident edges does not exceed α. α-MSTP consists in finding an α-ST with the least weight. We introduce two α-MSTP integer programming formulations, ℱ_xy^* and 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, ℱ_x^++, was obtained by: (i) lifting an existing set of inequalities in charge of enforcing α angular constraints and (ii) characterizing α-MSTP valid inequalities from the Stable Set polytope, a structure behind α-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 ℱ_x^++, BCFX^++ seems to outperform the other existing α-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.
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