五个选民无法诱导的锦标赛
Tournaments not inducible by five voters
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中文总结 AI 辅助
本研究通过定制搜索算法，将五个选民不可诱导锦标赛的最小阶数界限改进为13至23，并证明P23等实例不可诱导，同时给出可验证的机器证明。
中文摘要 AI 辅助
一个锦标赛 T 是 k-可诱导的，如果存在 k 个其顶点集上的线性序，使得对于 T 的每条弧 i → j，多数序都将 i 排在 j 之上。对于奇数 k，令 N(k) 为某个锦标赛不再是 k-可诱导的最小阶数。目前仅确切知道 N(3) = 8；对于 N(5)，之前的最佳界限为 12 ≤ N(5) ≤ 38，来自我们之前的论文 [2]，该论文还给出了第一个中等阶数的显式例子，即 Paley 锦标赛 P_{43}。结果：一个定制搜索算法改进了两端：13 ≤ N(5) ≤ 23。上界来自证明 P_{23} 不是 5-可诱导的，这是 Bachmeier 等人 [1] 报告无法判定的情况，他们的 SAT 求解器在累计六周内未终止；而我们的求解器在一台笔记本电脑上耗时 22 小时。下界来自对阶数 12 的分析。我们还证明了 P_{31} 不是 5-可诱导的，而 P_{19} 是 5-可诱导的，但不能以单位余量诱导，即不能通过每个弧恰好由三个选民对两个选民支持的剖面诱导。P_{19} 和 P_{23} 对于各自的性质都是弧临界的，而 P_{31} 和 P_{43} 不是顶点临界的：删除一个顶点后得到的锦标赛仍然不是 5-可诱导的。方法：搜索每次放置一个顶点，总是选择剩余选项最少的顶点，并传播后果。结合锦标赛的自同构，这在一台笔记本电脑上解决了整数规划或通用 SAT 求解器都无法解决的实例。对于 P_{19} 和 P_{23} 的反驳也得到了认证：搜索被分成独立的子问题，SAT 求解器为每个子问题生成机器可检查的证明，另一个程序重新检查每个证明。所有结果，在两个人检查的引理的前提下，均可从 https URL 重现。
英文摘要
A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.
发表机构
- MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London（帝国理工学院公共卫生学院全球传染病分析医学研究委员会中心）
- University of Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE（巴黎第九大学，PSL大学，法国国家科学研究中心UMR7243，LAMSADE）
机构由 AI 辅助整理，请以论文原文为准。
