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Hardness of Median and Center in the Ulam Metric

Authors Nick Fischer , Elazar Goldenberg , Mursalin Habib , Karthik C. S.

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Ulam distance
  • median
  • center
  • rank aggregation
  • fine-grained complexity

Metrics

Abstract

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings.
- Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21].
- Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive Õ(n² L)-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

Cite As Get BibTex

Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S.. Hardness of Median and Center in the Ulam Metric. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 111:1-111:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.111

Author Details

Nick Fischer
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
Elazar Goldenberg
  • The Academic College of Tel Aviv-Yaffo, Israel
Mursalin Habib
  • Rutgers University, Piscataway, NJ, USA
Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA

Funding

Part of this work was done while Mursalin Habib and Karthik C. S. were visiting INSAIT, Sofia University "St. Kliment Ohridski". Nick Fischer, Mursalin Habib, and Karthik C. S. were partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT as part of the Bulgarian National Roadmap for Research Infrastructure.
  • Habib, Mursalin: Supported by the National Science Foundation under Grants CCF-2313372, CCF-2422558, and CCF-2443697.
  • Karthik C. S.: Supported by the National Science Foundation under Grants CCF-2313372, CCF-2422558, and CCF-2443697.

Acknowledgements

The authors would like to thank the DIMACS Workshop on Efficient Algorithms for High Dimensional Metrics: New Tools for a special collaboration opportunity as well as Siam Habib and Antti Roeyskoe for helpful discussions.

References

  1. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 59-78. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.14.
  2. Amir Abboud, MohammadHossein Bateni, Vincent Cohen-Addad, Karthik C. S., and Saeed Seddighin. On complexity of 1-center in various metrics. In Nicole Megow and Adam D. Smith, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11-13, 2023, Atlanta, Georgia, USA, volume 275 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.1.
  3. 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.
  4. Amir Abboud, Nick Fischer, Elazar Goldenberg, Karthik C. S., and Ron Safier. Can you solve closest string faster than exhaustive search? 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 3:1-3:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.3.
  5. Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: Ranking and clustering. J. ACM, 55(5):23:1-23:27, 2008. URL: https://doi.org/10.1145/1411509.1411513.
  6. David Aldous and Persi Diaconis. Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bulletin of the American Mathematical Society, 36(4):413-432, 1999. Google Scholar
  7. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 2005-2039. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  8. Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, and Andreas Hofmeier. On the hardness of maximum rank aggregation problems. J. Discrete Algorithms, 31:2-13, 2015. URL: https://doi.org/10.1016/J.JDA.2014.10.002.
  9. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). SIAM J. Comput., 47(3):1087-1097, 2018. URL: https://doi.org/10.1137/15M1053128.
  10. Therese Biedl, Franz-Josef Brandenburg, and Xiaotie Deng. On the complexity of crossings in permutations. Discret. Math., 309(7):1813-1823, 2009. URL: https://doi.org/10.1016/J.DISC.2007.12.088.
  11. Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors. Handbook of Computational Social Choice. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781107446984.
  12. 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.
  13. Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 79-97. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.15.
  14. Boris Bukh, Karthik C. S., and Bhargav Narayanan. Applications of random algebraic constructions to hardness of approximation. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 237-244. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00032.
  15. 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.
  16. Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Approximating the median under the ulam metric. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 761-775. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.48.
  17. Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Clustering permutations: New techniques with streaming applications. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 31:1-31:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ITCS.2023.31.
  18. Diptarka Chakraborty, Syamantak Das, Arindam Khan, and Aditya Subramanian. Fair rank aggregation. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28 - December 9, 2022, 2022. URL: http://papers.nips.cc/paper_files/paper/2022/hash/974309ef51ebd89034adc64a57e304f2-Abstract-Conference.html.
  19. Diptarka Chakraborty, Kshitij Gajjar, and Agastya Vibhuti Jha. Approximating the center ranking under ulam. In Mikolaj Bojanczyk and Chandra Chekuri, editors, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2021, December 15-17, 2021, Virtual Conference, volume 213 of LIPIcs, pages 12:1-12:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2021.12.
  20. Graham Cormode, S. Muthukrishnan, and Süleyman Cenk Sahinalp. Permutation editing and matching via embeddings. In Fernando Orejas, Paul G. Spirakis, and Jan van Leeuwen, editors, Automata, Languages and Programming, 28th International Colloquium, ICALP 2001, Crete, Greece, July 8-12, 2001, Proceedings, volume 2076 of Lecture Notes in Computer Science, pages 481-492. Springer, 2001. URL: https://doi.org/10.1007/3-540-48224-5_40.
  21. Roee David, Karthik C. S., and Bundit Laekhanukit. On the complexity of closest pair via polar-pair of point-sets. SIAM J. Discret. Math., 33(1):509-527, 2019. URL: https://doi.org/10.1137/17M1128733.
  22. Colin de la Higuera and Francisco Casacuberta. Topology of strings: Median string is np-complete. Theor. Comput. Sci., 230(1-2):39-48, 2000. URL: https://doi.org/10.1016/S0304-3975(97)00240-5.
  23. Cynthia Dwork, Ravi Kumar, Moni Naor, and D. Sivakumar. Rank aggregation methods for the web. In Vincent Y. Shen, Nobuo Saito, Michael R. Lyu, and Mary Ellen Zurko, editors, Proceedings of the Tenth International World Wide Web Conference, WWW 10, Hong Kong, China, May 1-5, 2001, pages 613-622. ACM, 2001. URL: https://doi.org/10.1145/371920.372165.
  24. Ronald Fagin, Ravi Kumar, and D. Sivakumar. Efficient similarity search and classification via rank aggregation. In Alon Y. Halevy, Zachary G. Ives, and AnHai Doan, editors, Proceedings of the 2003 ACM SIGMOD International Conference on Management of Data, San Diego, California, USA, June 9-12, 2003, pages 301-312. ACM, 2003. URL: https://doi.org/10.1145/872757.872795.
  25. Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S. Hardness of median and center in the ulam metric. CoRR, abs/2504.16437, 2025. URL: https://doi.org/10.48550/arXiv.2504.16437.
  26. Moti Frances and Ami Litman. On covering problems of codes. Theory Comput. Syst., 30(2):113-119, 1997. URL: https://doi.org/10.1007/S002240000044.
  27. Nick Goldman, Paul Bertone, Siyuan Chen, Christophe Dessimoz, Emily M. LeProust, Botond Sipos, and Ewan Birney. Towards practical, high-capacity, low-maintenance information storage in synthesized DNA. Nat., 494(7435):77-80, 2013. URL: https://doi.org/10.1038/NATURE11875.
  28. Dan Gusfield. Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology. Cambridge University Press, 1997. URL: https://doi.org/10.1017/CBO9780511574931.
  29. Donna Harman. Ranking algorithms. In William B. Frakes and Ricardo A. Baeza-Yates, editors, Information Retrieval: Data Structures & Algorithms, pages 363-392. Prentice-Hall, 1992. Google Scholar
  30. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  31. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  32. Karthik C. S. and Pasin Manurangsi. On closest pair in euclidean metric: Monochromatic is as hard as bichromatic. Comb., 40(4):539-573, 2020. URL: https://doi.org/10.1007/S00493-019-4113-1.
  33. John G. Kemeny. Mathematics without numbers. Daedalus, 88(4):577-591, 1959. Google Scholar
  34. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 95-103. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250806.
  35. Teuvo Kohonen. Median strings. Pattern Recognit. Lett., 3(5):309-313, 1985. URL: https://doi.org/10.1016/0167-8655(85)90061-3.
  36. J. Kevin Lanctôt, Ming Li, Bin Ma, Shaojiu Wang, and Louxin Zhang. Distinguishing string selection problems. Inf. Comput., 185(1):41-55, 2003. URL: https://doi.org/10.1016/S0890-5401(03)00057-9.
  37. Xue Li, Xinlei Wang, and Guanghua Xiao. A comparative study of rank aggregation methods for partial and top ranked lists in genomic applications. Briefings in bioinformatics, 20(1):178-189, 2019. URL: https://doi.org/10.1093/BIB/BBX101.
  38. Yuting Liu, Tie-Yan Liu, Tao Qin, Zhiming Ma, and Hang Li. Supervised rank aggregation. In Carey L. Williamson, Mary Ellen Zurko, Peter F. Patel-Schneider, and Prashant J. Shenoy, editors, Proceedings of the 16th International Conference on World Wide Web, WWW 2007, Banff, Alberta, Canada, May 8-12, 2007, pages 481-490. ACM, 2007. URL: https://doi.org/10.1145/1242572.1242638.
  39. Carlos D. Martínez-Hinarejos, Alfons Juan, and Francisco Casacuberta. Use of median string for classification. In 15th International Conference on Pattern Recognition, ICPR'00, Barcelona, Spain, September 3-8, 2000, pages 2903-2906. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/ICPR.2000.906220.
  40. François Nicolas and Eric Rivals. Complexities of the centre and median string problems. In Ricardo A. Baeza-Yates, Edgar Chávez, and Maxime Crochemore, editors, Combinatorial Pattern Matching, 14th Annual Symposium, CPM 2003, Morelia, Michocán, Mexico, June 25-27, 2003, Proceedings, volume 2676 of Lecture Notes in Computer Science, pages 315-327. Springer, 2003. URL: https://doi.org/10.1007/3-540-44888-8_23.
  41. Samuel E. L. Oliveira, Victor Diniz, Anísio Lacerda, Luiz H. C. Merschmann, and Gisele L. Pappa. Is rank aggregation effective in recommender systems? an experimental analysis. ACM Trans. Intell. Syst. Technol., 11(2):16:1-16:26, 2020. URL: https://doi.org/10.1145/3365375.
  42. Pavel A. Pevzner. Computational molecular biology - an algorithmic approach. MIT Press, 2000. Google Scholar
  43. V. Y. Popov. Multiple genome rearrangement by swaps and by element duplications. Theor. Comput. Sci., 385(1-3):115-126, 2007. URL: https://doi.org/10.1016/J.TCS.2007.05.029.
  44. Cyrus Rashtchian, Konstantin Makarychev, Miklós Z. Rácz, Siena Ang, Djordje Jevdjic, Sergey Yekhanin, Luis Ceze, and Karin Strauss. Clustering billions of reads for DNA data storage. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett, editors, Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pages 3360-3371, 2017. URL: https://proceedings.neurips.cc/paper/2017/hash/ab7314887865c4265e896c6e209d1cd6-Abstract.html.
  45. Craige Schensted. Longest increasing and decreasing subsequences. Canadian Journal of mathematics, 13:179-191, 1961. Google Scholar
  46. Anke van Zuylen and David P. Williamson. Deterministic algorithms for rank aggregation and other ranking and clustering problems. In Christos Kaklamanis and Martin Skutella, editors, Approximation and Online Algorithms, 5th International Workshop, WAOA 2007, Eilat, Israel, October 11-12, 2007. Revised Papers, volume 4927 of Lecture Notes in Computer Science, pages 260-273. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-77918-6_21.
  47. Tiance Wang, John Sturm, Paul W. Cuff, and Sanjeev R. Kulkarni. Condorcet voting methods avoid the paradoxes of voting theory. In 50th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2012, Allerton Park & Retreat Center, Monticello, IL, USA, October 1-5, 2012, pages 201-203. IEEE, 2012. URL: https://doi.org/10.1109/ALLERTON.2012.6483218.
  48. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. URL: https://doi.org/10.1016/J.TCS.2005.09.023.
  49. H. Peyton Young. Condorcet’s theory of voting. American Political science review, 82(4):1231-1244, 1988. Google Scholar
  50. H. Peyton Young and Arthur Levenglick. A consistent extension of condorcet’s election principle. SIAM Journal on applied Mathematics, 35(2):285-300, 1978. Google Scholar
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