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# Geometric TSP on Sets

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LIPIcs.ISAAC.2023.6.pdf
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## Cite As

Henk Alkema and Mark de Berg. Geometric TSP on Sets. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.6

## Abstract

In One-of-a-Set TSP, also known as the Generalised TSP, the input is a collection 𝒫 : = {P_1, ..., P_r} of sets in a metric space and the goal is to compute a minimum-length tour that visits one element from each set. In the Euclidean variant of this problem, each P_i is a set of points in ℝ^d that is contained in a given hypercube H_i. We investigate how the complexity of Euclidean One-of-a-Set TSP depends on λ, the ply of the set ℋ := {H_1, ..., H_r} of hypercubes (The ply is the smallest λ such that every point in ℝ^d is in at most λ of the hypercubes). Furthermore, we show that the problem can be solved in 2^O(λ^{1/d} n^{1-1/d}) time, where n : = ∑_{i=1}^r |P_i| is the total number of points. Finally, we show that the problem cannot be solved in 2^o(n) time when λ = Θ(n), unless the Exponential Time Hypothesis (ETH) fails. In Rectilinear One-of-a-Cube TSP, the input is a set ℋ of hypercubes in ℝ^d and the goal is to compute a minimum-length rectilinear tour that visits every hypercube. We show that the problem can be solved in 2^O(λ^{1/d} n^{1-1/d} log n) time, where n is the number of hypercubes.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Computational geometry
##### Keywords
• Euclidean TSP
• TSP on Sets
• Rectilinear TSP
• TSP on Neighbourhoods

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