Improved formulations and branch-and-cut algorithms for the angular constrained minimum spanning tree problem

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 G define points the Euclidean plane while edges, line segments connecting them, are weighted by distance between their endpoints. A spanning tree \)-spanning \)-ST) if, for any \(i V\), smallest that encloses all corresponding to its i-incident edges does not exceed \). \)-MSTP consists finding \)-ST with least weight. In this work, we discuss families valid inequalities. One them lifting existing angular constraints found literature others come from Stable Set polytope, structure behind \)-STs disclosed here. We show despite being already satisfied previously strongest known formulation, \({\mathcal {F}}_{xy}\), these lifted capable strengthening another model so both become equally strong, at instances tested Inequalities polytope improve best Linear Programming Relaxation (LPRs) bounds about 1.6%, on average, hardest problem. Additionally, indicate how formulation {F}}_{xy}\) can be more effectively used Branch-and-cut (BC) algorithms, reducing number variables explicitly enforced integer constrained eliminating do change quality LPR bounds. Extensive computational experiments conducted here suggest combination ideas above allows us redefine performing almost entire spectrum \) values, exception easy instances, those \ge \frac{2\pi }{3}\). particular, ones (corresponding \{\frac{\pi }{2}, \frac{\pi }{3},\frac{2\pi }{5}\}\)) could solved proven optimality, BC algorithm suggested improves average CPU times factors up 5, average.