Makespan Trade-offs for Visiting Triangle Edges

Abstract

We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start from a point within a non-obtuse triangle ∆, 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 ∆ into 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 (∆) = max P ∈∆ R n (P )/R m (P ), and we prove that, over all non-obtuse triangles ∆: (i) R 1,3 (∆) ranges from 10 to 4, (ii) R 2,3 (∆) ranges from 2 to 2, and (iii) R 1,2 (∆) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle ∆ that maximize R n,m (∆), as well as we identify the triangles that determine all inf ∆ R n,m (∆) and sup ∆ R n,m (∆) over the set of non-obtuse triangles.