Ulam Median is NP-hard for Four Permutations

Mursalin Habib

Ulam Median is NP-hard for Four Permutations

Computational Complexity 2026-08-06 v1

Authors: Mursalin Habib

Abstract

We show that computing a median under the Ulam distance is NP-hard even when the input consists of exactly four permutations. Previously, NP-hardness was known only for an unbounded number of input permutations (Fischer et al., ESA '25). Our result is tight, since an Ulam median of three permutations can be computed in polynomial time (Chakraborty--Das--Krauthgamer, SODA '21).

Keywords

Ulam median NP-hardness permutations combinatorial optimization computational complexity

Cite

@article{arxiv.2608.05544,
  title  = {Ulam Median is NP-hard for Four Permutations},
  author = {Mursalin Habib},
  journal= {arXiv preprint arXiv:2608.05544},
  year   = {2026}
}