发表机构
Toyohashi University of Technology（丰桥技术科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过重排不等式为稳定婚姻问题的盾数建立二次下界，构造循环空心壳实例达到相应阻塞对数，并证明相关结构性质与局限。
AI 中文摘要
我们研究稳定婚姻问题的盾数，其定义为在所有大小为n的偏好分布上，完全匹配所能达到的最大阻塞对数量的最小值。我们利用重排不等式建立了普适的二次下界σ(n) ≥ ⌈n(n-2)/6⌉，并证明该下界是从均匀一阶矩论证中可获得的最强结论。我们进一步证明了尖锐的最小重排不等式，并构造了一个显式的循环空心壳实例，达到⌊(n-1)^2/4⌋个阻塞对。我们还建立了双团可实现性的充分霍尔型条件，证明了均匀平均和双参数线性分配目标的局限性，并为所有无条件结果提供了完整证明。计算机辅助验证仅用于明确识别的计算性陈述，并与数学证明清晰区分。
英文摘要
We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $σ(n) \ge \lceil n(n-2)/6 \rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\lfloor (n-1)^2/4 \rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.
Comments10 pages. Complete proofs are provided for all unconditional results. Computational verification is used only for explicitly identified computer-assisted statements
