Makespan Trade-offs for Visiting Triangle Edges
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We study a primitive vehicle routing-type problem in which a fleet of nunit speed robots start from a point within a non-obtuse triangle Δ, where n ∈{1,2,3}. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing Δinto regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan R_n(P) is determined, for n∈{1,2,3}. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define R_n,m (Δ)= max_P ∈Δ R_n(P)/R_m(P), and we prove that, over all non-obtuse triangles Δ: (i) R_1,3(Δ) ranges from √(10) to 4, (ii) R_2,3(Δ) ranges from √(2) to 2, and (iii) R_1,2(Δ) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle Δ that maximize R_n,m (Δ), as well as we identify the triangles that determine all inf_Δ R_n,m(Δ) and sup_Δ R_n,m(Δ) over the set of non-obtuse triangles.
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