Sublinear Algorithms for TSP via Path Covers

Authors Soheil Behnezhad , Mohammad Roghani , Aviad Rubinstein , Amin Saberi



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.19.pdf
  • Filesize: 1.22 MB
  • 16 pages

Document Identifiers

Author Details

Soheil Behnezhad
  • Northeastern University, Boston, MA, USA
Mohammad Roghani
  • Stanford University, CA, USA
Aviad Rubinstein
  • Stanford University, CA, USA
Amin Saberi
  • Stanford University, CA, USA

Cite AsGet BibTex

Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi. Sublinear Algorithms for TSP via Path Covers. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.19

Abstract

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, there is an algorithm that (1/2 - ε)-approximates the maximum path cover size of an n-vertex graph in Õ(n) time. This improves upon a (3/8-ε)-approximate Õ(n √n)-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an Õ(n) time algorithm that estimates the cost of (1,2)-TSP within a factor of (1.5+ε) which is an improvement over a folklore (1.75 + ε)-approximate Õ(n)-time algorithm, as well as a (1.625+ε)-approximate Õ(n√n)-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an Õ(n) algorithm that estimates the cost of graphic TSP within a factor of 1.83 which is an improvement over a 1.92-approximate Õ(n) time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to 1.66 using n^{2-Ω(1)} time. All of our Õ(n) time algorithms are information-theoretically time-optimal up to polylog n factors. Additionally, we show that our approximation guarantees for path cover and (1,2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sublinear Algorithms
  • Traveling Salesman Problem
  • Approximation Algorithm
  • (1
  • 2)-TSP
  • Graphic TSP

Metrics

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

References

  1. Anna Adamaszek, Matthias Mnich, and Katarzyna Paluch. New approximation algorithms for (1, 2)-tsp. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 9:1-9:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.ICALP.2018.9.
  2. Soheil Behnezhad. Time-Optimal Sublinear Algorithms for Matching and Vertex Cover. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 873-884. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00089.
  3. Soheil Behnezhad, Mohammad Roghani, and Aviad Rubinstein. Local computation algorithms for maximum matching: New lower bounds. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 2322-2335. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00143.
  4. Soheil Behnezhad, Mohammad Roghani, and Aviad Rubinstein. Sublinear time algorithms and complexity of approximate maximum matching. Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 267-280, 2023. URL: https://doi.org/10.1145/3564246.3585231.
  5. Soheil Behnezhad, Mohammad Roghani, and Aviad Rubinstein. Approximating maximum matching requires almost quadratic time. Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, Canada, To Appear, 2024. Google Scholar
  6. Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi. Beating greedy matching in sublinear time. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3900-3945. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH151.
  7. Piotr Berman and Marek Karpinski. 8/7-approximation algorithm for (1, 2)-tsp. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 641-648. ACM Press, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109627.
  8. Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak. Dynamic (1+ε)-approximate matching size in truly sublinear update time. 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1563-1588, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00095.
  9. Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak. Sublinear algorithms for (1.5+ε)-approximate matching. Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 254-266, 2023. URL: https://doi.org/10.1145/3564246.3585252.
  10. Yu Chen, Sampath Kannan, and Sanjeev Khanna. Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:19, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  11. Yu Chen, Sanjeev Khanna, and Zihan Tan. Sublinear algorithms and lower bounds for estimating MST and TSP cost in general metrics. 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, 261:37:1-37:16, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.37.
  12. Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group, 1976. Google Scholar
  13. Artur Czumaj and Christian Sohler. Estimating the weight of metric minimum spanning trees in sublinear-time. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 175-183, 2004. URL: https://doi.org/10.1145/1007352.1007386.
  14. Artur Czumaj and Christian Sohler. Estimating the weight of metric minimum spanning trees in sublinear time. SIAM J. Comput., 39(3):904-922, 2009. URL: https://doi.org/10.1137/060672121.
  15. Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. A randomized rounding approach to the traveling salesman problem. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 550-559. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.80.
  16. Anupam Gupta and Krzysztof Onak. Sublinear.info Open Problem 71: Metric TSP Cost Approximation, 2016. Available at URL: https://sublinear.info/index.php?title=Open_Problems:71.
  17. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 32-45. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451009.
  18. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  19. Marek Karpinski and Richard Schmied. On approximation lower bounds for TSP with bounded metrics. Electron. Colloquium Comput. Complex., page 8, 2012. URL: https://arxiv.org/abs/TR12-008.
  20. D. König. Über graphen und ihre anwendung auf determinantentheorie und mengenlehre. Mathematische Annalen, 77:453-465, 1916. URL: http://eudml.org/doc/158740.
  21. Matthias Mnich and Tobias Mömke. Improved integrality gap upper bounds for traveling salesperson problems with distances one and two. Eur. J. Oper. Res., 266(2):436-457, 2018. URL: https://doi.org/10.1016/J.EJOR.2017.09.036.
  22. Tobias Mömke and Ola Svensson. Approximating graphic TSP by matchings. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 560-569. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.56.
  23. Marcin Mucha. 13/9-approximation for graphic TSP. In 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 30-41. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPICS.STACS.2012.30.
  24. Christos H. Papadimitriou and Mihalis Yannakakis. The traveling salesman problem with distances one and two. Math. Oper. Res., 18(1):1-11, 1993. URL: https://doi.org/10.1287/MOOR.18.1.1.
  25. András Sebö and Jens Vygen. Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Comb., 34(5):597-629, 2014. URL: https://doi.org/10.1007/S00493-014-2960-3.
  26. Anatoliy I Serdyukov. O nekotorykh ekstremal’nykh obkhodakh v grafakh. Upravlyayemyye sistemy, 17:76-79, 1978. Google Scholar
  27. Yuichi Yoshida, Masaki Yamamoto, and Hiro Ito. An improved constant-time approximation algorithm for maximum matchings. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 225-234. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536447.