Kemeny Rank Aggregation for Three Voters
Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even ; three voters was the minimal open case, and is polynomial-time solvable.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Computational social choice
- Posed by
- Cynthia Dwork, Ravi Kumar, Moni Naor & D. Sivakumar
- Year posed
- 2001
- Years open
- 25y
- Solved
- 2026-07-28
- Model
- GPT-5.6 Sol Ultra, Claude Fable 5
- Vendor
- OpenAI / Anthropic
- Collaborators
- Dominik Peters
- Verification
- Lean-verified
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
GPT-5.6 Sol Ultra found the reduction from MAX CUT; Claude Fable 5 helped simplify parts of it. Together with earlier results, every fixed number of voters is now hard.
Verification
The reduction is Lean-checked, alongside an author-written arXiv preprint.
Source
arXiv:2607.25540 - Kemeny rank aggregation is NP-hard for three voters