Kemeny Rank Aggregation is NP-Hard for Three Voters

2026-07-28

AI Summary

This study resolves an open problem posed by Dwork et al. in 2001 concerning the computational complexity of Kemeny rank aggregation with only three voters. By constructing a polynomial-time reduction from the MAX CUT problem, the authors provide the first rigorous proof that Kemeny rank aggregation remains NP-complete even when the number of input rankings is exactly three. This result sharpens the known complexity boundary for the problem and establishes its intrinsic intractability at the smallest nontrivial number of voters, thereby completing the characterization of its computational hardness across all relevant input sizes.

Abstract

Rank aggregation is the task of combining $n$ input rankings (linear orders) of alternatives into a single output ranking. The Kemeny rank aggregation rule selects the output ranking that minimizes the total Kendall-tau distance to the input rankings, i.e., the total number of adjacent swaps that need to be performed across input rankings so that they are all equal to the output ranking. Dwork et al. (2001) proved that the problem of computing such a ranking is NP-complete for every even $n \ge 4$ and asked whether hardness holds even for $n = 3$. We give a hardness reduction from MAX CUT that proves the problem is NP-complete for $n = 3$. The reduction was found in July 2026 by GPT 5.6 Sol Ultra and simplified in part with help from Claude Fable 5.

Problem

• Kemeny Rank Aggregation
• NP-hard
• Rank Aggregation
• Kendall-tau distance
• Computational Complexity

Innovation

• Kemeny rank aggregation
• NP-completeness
• rank aggregation
• computational complexity
• MAX CUT reduction