Makespan Trade-offs for Visiting Triangle Edges
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912236183748608 |
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| author | Georgiou, Konstantinos Kundu, Somnath Pralat, Pawel |
| author_facet | Georgiou, Konstantinos Kundu, Somnath Pralat, Pawel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_01191 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Makespan Trade-offs for Visiting Triangle Edges Georgiou, Konstantinos Kundu, Somnath Pralat, Pawel Discrete Mathematics 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. |
| title | Makespan Trade-offs for Visiting Triangle Edges |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2105.01191 |