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Many Visits TSP Revisited

Authors Łukasz Kowalik , Shaohua Li, Wojciech Nadara, Marcin Smulewicz, Magnus Wahlström

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • many visits traveling salesman problem
  • exponential algorithm

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Abstract

We study the Many Visits TSP problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O*(5ⁿ). They also show a polynomial space algorithm running in time O(16^{n+o(n)}). In this work, we show three main results:  
- A randomized polynomial space algorithm in time O*(2^n D), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ε)-approximation in time O*(2ⁿε^{-1}). Improving the constant 2 in these results would be a major breakthrough, as it would result in improving the O*(2ⁿ)-time algorithm for Directed Hamiltonian Cycle, which is a 50 years old open problem. 
- A tight analysis of Berger et al.’s exponential space algorithm, resulting in an O*(4ⁿ) running time bound. 
- A new polynomial space algorithm, running in time O(7.88ⁿ).

Cite As Get BibTex

Łukasz Kowalik, Shaohua Li, Wojciech Nadara, Marcin Smulewicz, and Magnus Wahlström. Many Visits TSP Revisited. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.66

Author Details

Łukasz Kowalik
  • Institute of Informatics, University of Warsaw, Poland
Shaohua Li
  • Institute of Informatics, University of Warsaw, Poland
Wojciech Nadara
  • Institute of Informatics, University of Warsaw, Poland
Marcin Smulewicz
  • Institute of Informatics, University of Warsaw, Poland
Magnus Wahlström
  • Royal Holloway, University of London, UK

Funding

This workshop was supported by a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714704 (PI: Marcin Pilipczuk).
  • Kowalik, Łukasz: Supported by ERC Starting Grant TOTAL (Grant Agreement No 677651).
  • Li, Shaohua: Supported by ERC Starting Grant CUTACOMBS (Grant Agreement No 714704).
  • Nadara, Wojciech: Supported by ERC Starting Grant CUTACOMBS (Grant Agreement No 714704).
  • Smulewicz, Marcin: Supported by ERC Starting Grant TOTAL (Grant Agreement No 677651).

Acknowledgements

The research leading to the results presented in this paper was partially carried out during the Parameterized Algorithms Retreat of the University of Warsaw, PARUW 2020, held in Krynica-Zdrój in February 2020.

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