对于三个投票者，凯梅尼排序聚合是NP难的
Kemeny Rank Aggregation is NP-Hard for Three Voters
中文总结 AI 辅助
研究三个投票者的凯梅尼排序聚合问题，通过从最大割问题进行难度归约，证明该问题是NP完全问题，此归约由GPT 5.6 Sol Ultra发现并经克劳德寓言5部分简化。
中文摘要 AI 辅助
排序聚合是将n个替代方案的输入排名（线性顺序）组合成单个输出排名的任务。凯梅尼排序聚合规则选择使到输入排名的总肯德尔-陶距离最小的输出排名，即 across input rankings so that they are all equal to the output ranking。德沃尔克等人（2001年）证明，对于每个偶数n≥4，计算这样一个排名的问题是NP完全问题，并询问即使对于n = 3，难度是否仍然成立。我们通过从最大割问题进行难度归约，证明了该问题对于n = 3是NP完全问题。该归约由GPT 5.6 Sol Ultra于2026年7月发现，并在克劳德寓言5的帮助下进行了部分简化。
英文摘要
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.
