Makespan Trade-offs for Visiting Triangle Edges

TL;DR

This work studies a geometric vehicle routing problem where 1–3 unit-speed robots must visit all three edges of a non-obtuse triangle Δ, starting from a point P, with the goal of minimizing makespan. It introduces the R_n regions framework to partition Δ according to optimal visitation patterns and derives exact worst-case ratios $\mathcal{R}_{n,m}(Δ)$ for (1,3), (2,3), and (1,2) across all Δ, yielding tight infimum/supremum values: $\inf_Δ\mathcal{R}_{1,3}=\sqrt{10}$ and $\sup_Δ\mathcal{R}_{1,3}=4$, $\inf_Δ\mathcal{R}_{2,3}=\sqrt{2}$ and $\sup_Δ\mathcal{R}_{2,3}=2$, $\inf_Δ\mathcal{R}_{1,2}=\tfrac{5}{2}$ and $\sup_Δ\mathcal{R}_{1,2}=3$. The extremal cases are identified (equilateral, right isosceles, thin isosceles), with the incenter and the middle of the shortest altitude serving as key starting points for achieving the bounds. The results reveal a precise landscape of efficiency trade-offs for fleet sizes in a geometric VRP variant and introduce a versatile framework for analyzing optimal visitation structure via region decompositions and paraboloid/separator constructions. This framework has potential implications for broader geometric VRP problems and lower-bound analyses in related search/cover problems.

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 $Δ$, where $n \in \{1,2,3\}$. 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 $Δ$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 $n\in \{1,2,3\}$. 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 \in Δ} R_n(P)/R_m(P)$, and we prove that, over all non-obtuse triangles $Δ$: (i) $R_{1,3}(Δ)$ ranges from $\sqrt{10}$ to $4$, (ii) $R_{2,3}(Δ)$ ranges from $\sqrt{2}$ 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 $\inf_ΔR_{n,m}(Δ)$ and $\sup_ΔR_{n,m}(Δ)$ over the set of non-obtuse triangles.

Figures (23)

• Figure 1: Optimal trajectory for visiting $\{AB,AC\}$ for $\angle A\leq \pi/3$.
• Figure 2: Optimal trajectory for visiting $\{AB,AC\}$ for $\angle A\geq \pi/3$, starting outside the optimal bouncing subcone.
• Figure 3: Arbitrary non-obtuse $\triangle ABC$ shown with its LRD bounce indicator line (blue dotted line) and its LRD subopt indicator line (red dotted line).
• Figure 4: The $R_3$ regions of an arbitrary non-obtuse $\triangle ABC$. $AK, BL, CM$ are the angle bisectors of $\angle A, \angle B, \angle C$, respectively. Recall that the incenter $I$ is equidistant from all triangle edges.
• Figure 5: The case of $\angle B > \angle C$, in the proof of Lemma \ref{['lem: angle ABC R2 divisor']}.

Theorems & Definitions (68)

• proof
• Corollary 2.2
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• proof
• Lemma 3.3
• Lemma 4.1
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• Lemma 4.2
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• Lemma 4.3