{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-core-of-approval-based-committee","title":"The Core of Approval-Based Committee Elections with Few Seats","arxiv_id":"2501.18304","date":"2025-01-30","proceeding":null,"authors":["Dominik Peters"],"abstract":"In an approval-based committee election, the goal is to select a committee consisting of $k$ out of $m$ candidates, based on $n$ voters who each approve an arbitrary number of the candidates. The core of such an election consists of all committees that satisfy a certain stability property which implies proportional representation. In particular, committees in the core cannot be \"objected to\" by a coalition of voters who is underrepresented. The notion of the core was proposed in 2016, but it has remained an open problem whether it is always non-empty. We prove that core committees always exist when $k \\le 8$, for any number of candidates $m$ and any number of voters $n$, by showing that the Proportional Approval Voting (PAV) rule due to Thiele [1895] always satisfies the core when $k \\le 7$ and always selects at least one committee in the core when $k = 8$. We also develop an artificial rule based on recursive application of PAV, and use it to show that the core is non-empty whenever there are $m \\le 15$ candidates, for any committee size $k \\le m$ and any number of voters $n$. These results are obtained with the help of computer search using linear programs.","url_abs":"https://arxiv.org/abs/2501.18304v2","url_pdf":"https://arxiv.org/pdf/2501.18304v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-core-of-approval-based-committee","repo_url":"https://github.com/dominikpeters/core-few-candidates","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2501.18304","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}