The Angular Constrained Minimum Spanning Tree Problem ( $$\alpha $$ α -MSTP) is defined in terms of a complete undirected graph $$G=(V,E)$$ G = ( V , E ) and an angle $$\alpha \in (0,2\pi ]$$ α ∈ ( 0 , 2 π ] . 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 $$\alpha $$ α -spanning tree ( $$\alpha $$ α -ST) of G if, for any $$i \in V$$ i ∈ V , the smallest angle that encloses all line segments corresponding to its i -incident edges does not exceed $$\alpha $$ α . $$\alpha $$ α -MSTP consists in finding an $$\alpha $$ α -ST with the least weight. In this work, we discuss families of $$\alpha $$ α -MSTP valid inequalities. One of them is a lifting of existing angular constraints found in the literature and the others come from the Stable Set polytope, a structure behind $$\alpha $$ α -STs disclosed here. We show that despite being already satisfied by the previously strongest known formulation, $${\mathcal {F}}_{xy}$$ F xy , these lifted angular constraints are capable of strengthening another existing $$\alpha $$ α -MSTP model so that both become equally strong, at least for the instances tested here. Inequalities from the Stable Set polytope improve the best known Linear Programming Relaxation (LPRs) bounds by about 1.6%, on average, for the hardest instances of the problem. Additionally, we indicate how formulation $${\mathcal {F}}_{xy}$$ F xy can be more effectively used in Branch-and-cut (BC) algorithms, by reducing the number of variables explicitly enforced to be integer constrained and by eliminating constraints that do not change the quality of its LPR bounds. Extensive computational experiments conducted here suggest that the combination of the ideas above allows us to redefine the best performing $$\alpha $$ α -MSTP algorithms, for almost the entire spectrum of $$\alpha $$ α values, the exception being the easy instances, those with $$\alpha \ge \frac{2\pi }{3}$$ α ≥ 2 π 3 . In particular, for the hardest ones (corresponding to $$\alpha \in \{\frac{\pi }{2}, \frac{\pi }{3},\frac{2\pi }{5}\}$$ α ∈ { π 2 , π 3 , 2 π 5 } ) that could be solved to proven optimality, the best BC algorithm suggested here improves on average CPU times by factors of up to 5, on average.

中文翻译：

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角度约束最小生成树问题的改进公式和分支切割算法

其中之一是解除文献中发现的现有角度约束，其他来自稳定集多面体，这是此处公开的 $$\alpha $$ α -STs 背后的结构。我们表明，尽管先前已知最强的公式 $${\mathcal {F}}_{xy}$$ F xy 已经满足了，但这些提升的角度约束能够加强另一个现有的 $$\alpha $$ α - MSTP 模型使两者变得同样强大，至少对于这里测试的实例。对于最困难的问题实例，来自稳定集多面体的不等式平均将最著名的线性规划松弛 (LPR) 边界提高了约 1.6%。此外，我们指出公式 $${\mathcal {F}}_{xy}$$ F xy 如何更有效地用于分支切割（BC）算法，通过减少明确强制为整数约束的变量的数量，并消除不改变其 LPR 边界质量的约束。这里进行的大量计算实验表明，结合上述想法，我们可以重新定义性能最佳的 $$\alpha $$ α -MSTP 算法，几乎适用于整个 $$\alpha $$ α 值范围，除了简单的例子，那些 $$\alpha \ge \frac{2\pi }{3}$$ α ≥ 2 π 3 。特别是对于最难的（对应于 $$\alpha \in \{\frac{\pi }{2}, \frac{\pi }{3},\frac{2\pi }{5}\} $$ α ∈ { π 2 , π 3 , 2 π 5 } ) 可以解决到已证明的最优性，这里建议的最佳 BC 算法平均将 CPU 时间提高了 5 倍。