The Complexity of Kemeny Aggregation with Three Rankings

2026-07-30

AI Summary

This study investigates the computational complexity of the Kemeny aggregation problem when restricted to exactly three unweighted input rankings. By constructing specific ternary ranking instances, it establishes for the first time that computing the Kemeny score in this setting is NP-complete, while winner determination and optimal solution verification are respectively Θ₂^p-complete and coNP-complete. The analysis combines complexity-theoretic reductions, structural properties of tournament graphs, and statistical ranking techniques under the Mallows model to uncover phase transitions in complexity driven by the dimensionality of majority tournaments and support thresholds. Key contributions include a complete complexity classification for fixed small input sizes, extensions to Slater consensus and median permutation problems, and an affine characterization—along with uniqueness conditions—of optimal solutions under equidistant triples derived from six-fold replicated instances.

Abstract

The Kemeny rule aggregates rankings by minimizing their total Kendall-tau distance from an aggregate order. We prove that Kemeny Score is NP-complete for exactly three unweighted rankings, even when every candidate pair is split $2$-to-$1$. On the same profiles, the winner, unique-winner, and possible- and necessary-precedence problems are $Θ_2^p$-complete, while recognizing a Kemeny-optimal or uniquely Kemeny-optimal aggregate is coNP-complete. The hard instances induce tournaments of majority dimension exactly $3$. The reduction also determines the exact maximum-cut value from the optimal Kemeny score and recovers a maximum cut from any Kemeny-optimal aggregate. For every fixed $q\geq3$ and $\lceil q/2\rceil\leq s\leq q$, minimum pairwise support $s$ yields a sharp dichotomy: the score problem is NP-complete, the winner and precedence problems are $Θ_2^p$-complete, and the recognition problems are coNP-complete when $3s\leq2q$; for $3s>2q$, the majority tournament is transitive and its unique topological order is the unique Kemeny-optimal aggregate. Exact support $s$ suffices in the hard case when $s>q/2$, and supports in ${s,s+1}$ suffice when $s=q/2$. These results give complete fixed-profile-size classifications and transfer to Slater orders, permutation medians, and maximum-likelihood central rankings in the Mallows model. Finally, a six-copy construction proves NP-completeness of both Kemeny Score and Kendall--Tau Center for three pairwise-equidistant rankings that still split every pair $2$-to-$1$. For $N$ output candidates, their common distance is $\frac23\binom N2$, the largest possible for an equidistant triple. The construction gives affine formulas for both optimal values, characterizes all Kemeny-optimal output orders, and shows that the output has a unique Kemeny-optimal order and a unique center exactly when the input has a unique Kemeny-optimal order.

Problem

• Kemeny aggregation
• computational complexity
• Kendall-tau distance
• NP-completeness
• rank aggregation

Innovation

• Kemeny aggregation
• computational complexity
• NP-completeness
• majority tournaments
• maximum-cut reduction