Document Open Access Logo

From Donkeys to Kings in Tournaments

Authors Amir Abboud, Tomer Grossman, Moni Naor , Tomer Solomon

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.3.pdf
  • Filesize: 0.84 MB
  • 14 pages

Document Identifiers

Author Details

Amir Abboud
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Tomer Grossman
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Moni Naor
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Tomer Solomon
  • Tel Aviv University, Israel

Funding

  • Abboud, Amir: This work is part of the project CONJEXITY, which has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation program (grant agreement No. 101078482). Supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.
  • Naor, Moni: Incumbent of the Judith Kleeman Professorial Chair. Supported in part by grants from the Israel Science Foundation (no.2686/20), by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness and by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center.
  • Solomon, Tomer: This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

Cite AsGet BibTex

Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon. From Donkeys to Kings in Tournaments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.3

Abstract

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 3 kings is O(n), and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes O(n^{3/2}) queries). On the other hand, we show that finding k ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (the time for matrix multiplication). We provide evidence that this is optimal for large k by suggesting a fine-grained reduction from a variant of the triangle detection problem.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Tournament Graphs
  • Kings
  • Query Complexity
  • Fine Grained Complexity

Metrics

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

References

  1. Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Scheduling lower bounds via AND subset sum. J. Comput. Syst. Sci., 127:29-40, 2022. URL: https://doi.org/10.1016/J.JCSS.2022.01.005.
  2. Amir Abboud, Mina Dalirrooyfard, Ray Li, and Virginia Vassilevska Williams. On diameter approximation in directed graphs. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 2:1-2:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.2.
  3. Amir Abboud, Nick Fischer, Zander Kelley, Shachar Lovett, and Raghu Meka. New graph decompositions and combinatorial boolean matrix multiplication algorithms. Electron. Colloquium Comput. Complex., TR23-180, 2023. URL: https://arxiv.org/abs/TR23-180.
  4. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.53.
  5. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH28.
  6. Miklós Ajtai, Vitaly Feldman, Avinatan Hassidim, and Jelani Nelson. Sorting and selection with imprecise comparisons. ACM Trans. Algorithms, 12(2):19:1-19:19, 2016. URL: https://doi.org/10.1145/2701427.
  7. R. Balasubramanian, Venkatesh Raman, and G. Srinivasaragavan. Finding scores in tournaments. J. Algorithms, 24(2):380-394, 1997. URL: https://doi.org/10.1006/JAGM.1997.0865.
  8. Amotz Bar-Noy and Joseph Naor. Sorting, minimal feedback sets, and hamilton paths in tournaments. SIAM J. Discret. Math., 3(1):7-20, 1990. URL: https://doi.org/10.1137/0403002.
  9. Arindam Biswas, Varunkumar Jayapaul, Venkatesh Raman, and Srinivasa Rao Satti. Finding kings in tournaments. Discret. Appl. Math., 322:240-252, 2022. URL: https://doi.org/10.1016/J.DAM.2022.08.014.
  10. Karl Bringmann and Bhaskar Ray Chaudhury. Polyline simplification has cubic complexity. J. Comput. Geom., 11(2):94-130, 2020. URL: https://doi.org/10.20382/JOCG.V11I2A5.
  11. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci., 288(1):21-43, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00144-X.
  12. Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 261-270. ACM, 2016. URL: https://doi.org/10.1145/2840728.2840746.
  13. Dishant Goyal, Varunkumar Jayapaul, and Venkatesh Raman. Elusiveness of finding degrees. Discret. Appl. Math., 286:128-139, 2020. URL: https://doi.org/10.1016/J.DAM.2019.06.009.
  14. Tomer Grossman, Ilan Komargodski, and Moni Naor. Instance complexity and unlabeled certificates in the decision tree model. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 56:1-56:38. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ITCS.2020.56.
  15. Zhiyi Huang, Yaowei Long, Thatchaphol Saranurak, and Benyu Wang. Tight conditional lower bounds for vertex connectivity problems. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 1384-1395. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585223.
  16. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers. Springer Berlin Heidelberg, 2012. Google Scholar
  17. H. G. Landau. On dominance relations and the structure of animal societies: III the condition for a score structure. The Bulletin of Mathematical Biophysics, 15:143-148, 1953. Google Scholar
  18. Xiaoyun Lu, Da-Wei Wang, and C. K. Wong. On the bounded domination number of tournaments. Discret. Math., 220(1-3):257-261, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00029-7.
  19. Arnab Maiti and Palash Dey. Query complexity of tournament solutions. Theor. Comput. Sci., 991:114422, 2024. Google Scholar
  20. Nikhil S. Mande, Manaswi Paraashar, and Nitin Saurabh. Randomized and quantum query complexities of finding a king in a tournament. In Patricia Bouyer and Srikanth Srinivasan, editors, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2023, December 18-20, 2023, IIIT Hyderabad, Telangana, India, volume 284 of LIPIcs, pages 30:1-30:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2023.30.
  21. Stephen B. Maurer. The king chicken theorems. Mathematics Magazine, 53(2):67-80, 1980. URL: https://doi.org/10.1080/0025570X.1980.11976831.
  22. Nimrod Megiddo and Uzi Vishkin. On finding a minimum dominating set in a tournament. Theor. Comput. Sci., 61:307-316, 1988. URL: https://doi.org/10.1016/0304-3975(88)90131-4.
  23. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
  24. Ariel D. Procaccia. A note on the query complexity of the condorcet winner problem. Inf. Process. Lett., 108(6):390-393, 2008. URL: https://doi.org/10.1016/J.IPL.2008.07.012.
  25. Jian Shen, Li Sheng, and Jie Wu. Searching for sorted sequences of kings in tournaments. SIAM J. Comput., 32(5):1201-1209, 2003. URL: https://doi.org/10.1137/S0097539702410053.
  26. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. URL: https://doi.org/10.1145/3186893.
  27. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega. CoRR, abs/2307.07970, 2023. URL: https://doi.org/10.48550/arXiv.2307.07970.
  28. Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In 18th Annual Symposium on Foundations of Computer Science, FOCS, pages 222-227. IEEE Computer Society, 1977. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact