Modeling and solving the angular constrained minimum spanning tree problem

Abstract

Assume one is given an angle α ∈ (0, 2π] and a complete undirected graph G=(V,E). The vertices in V represent points in the Euclidean plane. The edges in E represent the line segments between these points, with edge weights equal to segment lengths. Spanning trees of G are called α-spanning trees (α-STs) if, for any i ∈ V, the smallest angle that encloses all line segments corresponding to its i-incident edges does not exceed α. The Angular Constrained Minimum Spanning Tree Problem (α-MSTP) seeks an α-ST with the least weight. The problem arises in the design of wireless communication networks operating under directional antennas. We propose two α-MSTP formulations. One, Fx requiring, in principle, O(2|V|) inequalities to model the angular constraints (α-AC). For α ∈ (0, π), however, we show that just O(|V|3) of them suffice to attain not only a formulation but also the same Linear Programming relaxation (LPR) bound as the full blown model. The other formulation, Fxy, enlarges the set of Fx variables but manages to model α-AC, compactly. Furthermore, Fxy LPR bounds are proven to dominate their Fx counterparts. That dominance, however, is empirically shown to reduce as α increases. Finally, exact Branch-and-Cut algorithms implemented for the two formulations are shown, empirically, to exchange roles as top performer, throughout the [0, 2π) range of α values.