2026-08-18T03:52:57Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:24580 2025-12-16T14:01:06Z ddc:004 open_access

Hardness of Median and Center in the Ulam Metric Fischer, Nick Goldenberg, Elazar Habib, Mursalin Karthik C. S. Ulam distance median center rank aggregation fine-grained complexity 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Nick Fischer and Elazar Goldenberg and Mursalin Habib and Karthik C. S. 2025 Is Part Of LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2025.111 urn:nbn:de:0030-drops-245809 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.111 eng https://creativecommons.org/licenses/by/4.0/legalcode