Geometric TSP on Sets

Authors Henk Alkema, Mark de Berg



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Henk Alkema
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

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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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References

  1. Binay Bhattacharya, Ante Ćustić, Akbar Rafiey, Arash Rafiey, and Vladyslav Sokol. Approximation algorithms for generalized mst and TSP in grid clusters. In Zaixin Lu, Donghyun Kim, Weili Wu, Wei Li, and Ding-Zhu Du, editors, Combinatorial Optimization and Applications, pages 110-125, Cham, 2015. Springer International Publishing. Google Scholar
  2. T.-H. Hubert Chan and Shaofeng H.-C. Jiang. Reducing curse of dimensionality: Improved PTAS for TSP (with neighborhoods) in doubling metrics. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 754-765. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch54.
  3. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, and Sudeshna Kolay. An ETH-tight exact algorithm for euclidean TSP. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 450-461, 2018. URL: https://doi.org/10.1109/FOCS.2018.00050.
  4. Mark de Berg, Joachim Gudmundsson, Matthew J. Katz, Christos Levcopoulos, Mark H. Overmars, and A. Frank van der Stappen. TSP with neighborhoods of varying size. Journal of Algorithms, 57(1):22-36, 2005. URL: https://doi.org/10.1016/j.jalgor.2005.01.010.
  5. Moshe Dror and James B. Orlin. Combinatorial optimization with explicit delineation of the ground set by a collection of subsets. SIAM Journal on Discrete Mathematics, 21(4):1019-1034, 2008. URL: https://doi.org/10.1137/050636589.
  6. M. R. Garey, Ronald L. Graham, and David S. Johnson. Some NP-complete geometric problems. In STOC, pages 10-22. ACM, 1976. Google Scholar
  7. Gregory Gutin and Abraham P. Punnen. The Traveling Salesman Problem and Its Variations. Springer, 2006. Google Scholar
  8. R. Z. Hwang, R. C. Chang, and Richard C. T. Lee. The searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica, 9(4):398-423, 1993. URL: https://doi.org/10.1007/BF01228511.
  9. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  10. Viggo Kann. On the approximability of NP-complete optimization problems. PhD thesis, Royal Institute of Technology Stockholm, 1992. Google Scholar
  11. Michael Khachay and Katherine Neznakhina. Approximation algorithms for generalized TSP in grid clusters. CEUR Workshop Proceedings, 1623:39-48, 2016. Google Scholar
  12. Michael Khachay and Katherine Neznakhina. Complexity and approximability of the euclidean generalized traveling salesman problem in grid clusters. Annals of Mathematics and Artificial Intelligence, 88(1):53-69, 2020. URL: https://doi.org/10.1007/s10472-019-09626-w.
  13. Joseph S. B. Mitchell. A PTAS for TSP with neighborhoods among fat regions in the plane. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pages 11-18, 2007. Google Scholar
  14. Joseph S.B. Mitchell. A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane. In Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, pages 183-191, 2010. URL: https://doi.org/10.1145/1810959.1810992.
  15. Christos H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theor. Comput. Sci., 4(3):237-244, 1977. Google Scholar
  16. Warren D. Smith and Nicholas C. Wormald. Geometric separator theorems & applications. In FOCS, pages 232-243. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743449.
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