Ulam Median is NP-hard for Four Permutations

2026-08-05

AI Summary

This study investigates the computational complexity of finding a median permutation under the Ulam distance. By constructing a strict reduction from a known NP-hard problem, it establishes that the problem remains NP-hard even when the input consists of only four permutations. This result significantly tightens the previous understanding of the problem’s hardness, which was limited to unbounded input sizes, and bridges the theoretical gap between the case of three permutations—solvable in polynomial time—and larger instances. Consequently, the work demonstrates that four permutations constitute the smallest input size for which the median permutation problem under the Ulam distance is provably NP-hard, thereby establishing the tightness of this complexity threshold.

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).

Problem

• Ulam median
• NP-hard
• permutations
• computational complexity

Innovation

• Ulam distance
• median permutation
• NP-hardness
• computational complexity
• combinatorial optimization