AI 中文总结
该研究证明三个未加权排序的Kemeny聚合相关问题的复杂性，给出固定剖面大小的完整分类，结果可迁移至Slater序等，还构造六副本证明特定Kendall-Tau Center问题的NP完全性。
AI 中文摘要
Kemeny规则通过最小化各排序与聚合序之间的总Kendall-tau距离来聚合排序。我们证明，对于恰好三个未加权排序，即使每个候选对以2比1分裂，Kemeny Score仍是NP完全的。在相同的偏好剖面中，胜者、唯一胜者、可能优先和必要优先问题是Θ₂^p完全的，而识别Kemeny最优或唯一Kemeny最优聚合是coNP完全的。困难实例诱导出多数维数恰好为3的竞赛图。该归约还从最优Kemeny得分确定了最大割的精确值，并从任何Kemeny最优聚合中恢复出最大割。对于每个固定的q≥3和⌈q/2⌉≤s≤q，最小成对支持度s会产生尖锐二分：当3s≤2q时，得分问题是NP完全的，胜者和优先问题是Θ₂^p完全的，识别问题是coNP完全的；当3s>2q时，多数竞赛图是传递的，其唯一拓扑序是唯一的Kemeny最优聚合。当s>q/2时，困难情形仅需精确支持度s，当s=q/2时，支持度{s,s+1}即可。这些结果给出了固定偏好剖面大小的完整分类，并可迁移到Slater序、排列中位数以及Mallows模型中的最大似然中心排序。最后，一个六副本构造证明，对于三个两两等距且仍以2比1分裂每对的排序，Kemeny Score和Kendall-Tau Center均是NP完全的。对于N个输出候选，它们的共同距离为(2/3)·C(N,2)，是等距三元组的最大可能值。该构造为两个最优值给出了仿射公式，刻画了所有Kemeny最优输出序，并表明当输入具有唯一Kemeny最优序时，输出具有唯一Kemeny最优序和唯一中心。
英文摘要
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.
