Popular branchings and their dual certificates

• 成果类型: Article
• 署名作者: Kavitha, Telikepalli; Kiraly, Tamas; Matuschke, Jannik; Schlotter, Ildiko; Schmidt-Kraepelin, Ulrike
• 署名单位: Tata Institute of Fundamental Research (TIFR); Eotvos Lorand University; KU Leuven; HUN-REN; HUN-REN Centre for Economic & Regional Studies; Budapest University of Technology & Economics; Technical University of Berlin
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01659-6
• 发表日期: 2022
• 页码: 567-595
• 关键词: matchings; marriage

摘要

Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that approximately popular branchings always exist.