Improved Guarantees for the a Priori TSP

Authors Jannis Blauth , Meike Neuwohner , Luise Puhlmann , Jens Vygen



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2023.14.pdf
  • Filesize: 0.82 MB
  • 16 pages

Document Identifiers

Author Details

Jannis Blauth
  • Research Inst. for Discrete Mathematics, Hausdorff Center for Math., University of Bonn, Germany
Meike Neuwohner
  • Research Inst. for Discrete Mathematics, Hausdorff Center for Math., University of Bonn, Germany
Luise Puhlmann
  • Research Inst. for Discrete Mathematics, Hausdorff Center for Math., University of Bonn, Germany
Jens Vygen
  • Research Inst. for Discrete Mathematics, Hausdorff Center for Math., University of Bonn, Germany

Cite As Get BibTex

Jannis Blauth, Meike Neuwohner, Luise Puhlmann, and Jens Vygen. Improved Guarantees for the a Priori TSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.14

Abstract

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space (V,c) and an activation probability p(v) for each customer v ∈ V. We ask for a TSP tour T for V that minimizes the expected length after cutting T short by skipping the inactive customers. 
All known approximation algorithms select a nonempty subset S of the customers and construct a master route solution, consisting of a TSP tour for S and two edges connecting every customer v ∈ V⧵S to a nearest customer in S. 
We address the following questions. If we randomly sample the subset S, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • A priori TSP
  • random sampling
  • stochastic combinatorial optimization

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Mohamed Abdellahi Amar, Walid Khaznaji, and Monia Bellalouna. An exact resolution for the probabilistic traveling salesman problem under the a priori strategy. Procedia Computer Science, 108:1414-1423, 2017. URL: https://doi.org/10.1016/j.procs.2017.05.068.
  2. Dimitris Bertsimas. Probabilistic combinatorial optimization problems. PhD thesis, Massachusetts Institute of Technology, 1988. URL: https://dspace.mit.edu/handle/1721.1/14386.
  3. Dimitris J. Bertsimas, Patrick Jaillet, and Amedeo R. Odoni. A priori optimization. Operations Research, 38(6):1019-1033, 1990. URL: https://doi.org/10.1287/opre.38.6.1019.
  4. Jannis Blauth, Meike Neuwohner, Luise Puhlmann, and Jens Vygen. Improved guarantees for the a priori TSP, 2023. URL: https://doi.org/10.48550/arXiv.2309.10663.
  5. Neill E. Bowler, Thomas M. A. Fink, and Robin C. Ball. Characterization of the probabilistic traveling salesman problem. Phys. Rev. E, 68:036703, 2003. URL: https://doi.org/10.1103/PhysRevE.68.036703.
  6. Martijn van Ee, Leo van Iersel, Teun Janssen, and René Sitters. A priori TSP in the scenario model. Discrete Applied Mathematics, 250:331-341, 2018. URL: https://doi.org/10.1016/j.dam.2018.04.002.
  7. Martijn van Ee and René Sitters. The a priori traveling repairman problem. Algorithmica, 80(10):2818-2833, 2018. URL: https://doi.org/10.1007/s00453-017-0351-z.
  8. Friedrich Eisenbrand, Fabrizio Grandoni, Thomas Rothvoß, and Guido Schäfer. Connected facility location via random facility sampling and core detouring. Journal of Computer and System Sciences, 76(8):709-726, 2010. URL: https://doi.org/10.1016/j.jcss.2010.02.001.
  9. Finn Fernstrøm and Teresa Anna Steiner. A constant approximation algorithm for the uniform a priori capacitated vehicle routing problem with unit demands. Information Processing Letters, 159-160:105960, 2020. URL: https://doi.org/10.1016/j.ipl.2020.105960.
  10. Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi. Robust algorithms for TSP and Steiner Tree. ACM Trans. Algorithms, 19(2), 2023. URL: https://doi.org/10.1145/3570957.
  11. Naveen Garg, Anupam Gupta, Stefano Leonardi, and Piotr Sankowski. Stochastic analyses for online combinatorial optimization problems. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 942-951. SIAM, 2008. URL: https://dl.acm.org/doi/10.5555/1347082.1347185.
  12. Michel Goemans and Jon Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82(1):111-124, 1998. URL: https://doi.org/10.1007/BF01585867.
  13. Igor Gorodezky, Robert D. Kleinberg, David B. Shmoys, and Gwen Spencer. Improved lower bounds for the universal and a priori TSP. In Maria Serna, Ronen Shaltiel, Klaus Jansen, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 178-191. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15369-3_14.
  14. Anupam Gupta, Amit Kumar, Martin Pál, and Tim Roughgarden. Approximation via cost sharing: Simpler and better approximation algorithms for network design. J. ACM, 54(3), 2007. URL: https://doi.org/10.1145/1236457.1236458.
  15. Anupam Gupta, Amit Kumar, and Tim Roughgarden. Simpler and better approximation algorithms for network design. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 365-372. ACM, 2003. URL: https://doi.org/10.1145/780542.780597.
  16. Anupam Gupta, Martin Pál, R. Ravi, and Amitabh Sinha. Boosted sampling: Approximation algorithms for stochastic optimization. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pages 417-426. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007419.
  17. Patrick Jaillet. Probabilistic traveling salesman problems. PhD thesis, Massachusetts Institute of Technology, 1985. URL: https://dspace.mit.edu/handle/1721.1/15231.
  18. Patrick Jaillet. A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research, 36(6):929-936, 1988. URL: https://doi.org/10.1287/opre.36.6.929.
  19. Kamal Jain, Mohammad Mahdian, Evangelos Markakis, Amin Saberi, and Vijay V. Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp. J. ACM, 50(6):795-824, 2003. URL: https://doi.org/10.1145/950620.950621.
  20. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In Proceedings of the 53rd Annual ACM Symposium on Theory of Computing, pages 32-45. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451009.
  21. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A deterministic better-than-3/2 approximation algorithm for metric TSP. In Alberto Del Pia and Volker Kaibel, editors, Integer Programming and Combinatorial Optimization, pages 261-274. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32726-1_19.
  22. Fatemeh Navidi, Inge Li Gørtz, and Viswanath Nagarajan. Approximation algorithms for the a priori traveling repairman. Operations Research Letters, 48(5):599-606, 2020. URL: https://doi.org/10.1016/j.orl.2020.07.009.
  23. Christos H. Papadimitriou and Mihalis Yannakakis. The traveling salesman problem with distances one and two. Mathematics of Operations Research, 18(1):1-11, 1993. URL: https://doi.org/10.1287/moor.18.1.1.
  24. Frans Schalekamp and David B. Shmoys. Algorithms for the universal and a priori TSP. Operations Research Letters, 36(1):1-3, 2008. URL: https://doi.org/10.1016/j.orl.2007.04.009.
  25. David Shmoys and Kunal Talwar. A constant approximation algorithm for the a priori traveling salesman problem. In Andrea Lodi, Alessandro Panconesi, and Giovanni Rinaldi, editors, Integer Programming and Combinatorial Optimization, pages 331-343. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-68891-4_23.
  26. Alejandro Toriello, William B. Haskell, and Michael Poremba. A dynamic traveling salesman problem with stochastic arc costs. Operations Research, 62(5):1107-1125, 2014. URL: https://doi.org/10.1287/opre.2014.1301.
  27. Laurence A. Wolsey. Heuristic analysis, linear programming and branch and bound. Mathematical Programming Study, 13:121-134, 1980. URL: https://doi.org/10.1007/BFb0120913.
  28. Anke van Zuylen. Deterministic sampling algorithms for network design. Algorithmica, 60(1):110-151, 2011. URL: https://doi.org/10.1007/s00453-009-9344-x.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail