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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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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

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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.

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