带有比较查询的带见证者的公平分配
Witness-Certified Fair Division with Comparison Queries
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中文总结 AI 辅助
该研究针对仅通过商品束序数比较访问估值的不可分商品公平分配问题，提出保留证书的缩放框架，实现高效EF1分配，优化相关算法的查询复杂度并研究证书结构。
中文摘要 AI 辅助
我们研究不可分商品的公平分配问题，其中智能体的估值仅通过对商品束的序数比较来访问，且存在任意平局打破规则。在该模型下，甚至无法判定给定分配是否为无嫉妒至多一件商品（EF1）分配，这表明显式公平见证证书是一种自然的算法对象。我们的主要贡献是一种保留证书的缩放框架，该框架递归收缩商品、求解规模更小的实例，再扩展解并修复显式的嫉妒消除见证证书。对于任意相同的单调估值，该框架可为n个智能体和m件商品生成经认证的EF1分配，使用O(n log n log(m/n))次比较查询，其查询复杂度与通信下界Ω(n log(m/n))的比值在O(log n)因子内。对于相同的加法估值，我们还在相同查询复杂度下获得了1/2-最大最小份额（MMS）保证。对于非相同的加法估值，我们利用我们的EF1+1/2-MMS算法，加速了现有的基于匹配的PROP1+1/2-MMS框架，将查询复杂度从O(n⁴ log m)提升至O(n³ log m)。最后，我们通过k-见证EF1（一种介于EF1和EFX之间的层级结构）研究此类证书的结构。
英文摘要
We study fair division of indivisible goods when agents' valuations are accessed only through ordinal comparisons between bundles, with arbitrary tie-breaking. In this model, even deciding whether a given allocation is envy-free up to one good (EF1) can be impossible. This suggests explicit fairness certificates as a natural algorithmic object. Our main contribution is a certificate-preserving scaling framework, which recursively contracts goods, solves a smaller instance, and expands the solution while repairing an explicit envy-eliminating witness certificate. For arbitrary identical monotone valuations, this yields a certified EF1 allocation for $n$ agents and $m$ goods using $O(n \log n \log(m/n))$ comparison queries, within an $O(\log n)$ factor of the $Ω(n \log (m/n))$ communication lower bound. For identical additive valuations, we additionally obtain a $1/2$-MMS guarantee within the same query complexity. For non-identical additive valuations, exploiting our EF1+$1/2$-MMS algorithm, we accelerate the existing matching-based PROP1+$1/2$-MMS framework, improving the query complexity from $O(n^4\log m)$ to $O(n^3\log m)$. Finally, we study the structure of such certificates through $k$-witness EF1, a hierarchy between EF1 and EFX.
