剩余最大最小份额：精确有限智能体边界、稀疏极值器与阈值割
Residual Maximin Share: Exact Finite-Agent Frontier, Sparse Extremizers, and Threshold Cuts
- University of Chinese Academy of Sciences（中国科学院大学）
- Chinese Academy of Sciences（中国科学院）
- Huawei Technologies Co., Ltd.（华为技术有限公司）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明剩余最大最小份额（RMMS）与经典最大最小份额（MMS）的有限智能体下界精确，构造线性支撑的三值极值实例，建立RMMS的结构特征，为基于份额的单独分割算法的公平保证设定严格限制。
AI中文摘要：
剩余最大最小份额（RMMS）是动态分配过程中仍可保证的最大份额阈值，即便之前已分配的低价值束从物品池中移除后依然成立。对于加法估值，近期密度平衡分析建立了比较RMMS与经典最大最小份额（MMS）的有限智能体下界。本文证明这些有限智能体下界是精确的：若$d_n$表示不超过$n$的最大奇数，最坏情况下的比值满足$\frac{\text{RMMS}(M,v,n)}{\text{MMS}(M,v,n)}$的下确界为$\frac{2d_n}{3d_n-1}$，因此精确加法边界形成连续奇偶平台，并单调收敛至2/3。随后研究极值实例的组合结构：朴素实例需要$\theta(n^2)$个物品，而构造的三值族仅需线性支撑即可达到精确边界，奇数$n$对应$(5n-3)/2$个物品，偶数$n$对应$(5n-4)/2$个物品；其低值块支持两个精确划分，同时可验证MMS基准与剩余障碍，将这两个对偶划分为二分运输图建模后，证明该块达到绝对最小支撑$q+d-1=3q$，最小支撑下任意二值填充在重标记下唯一刚性。最后建立RMMS的结构特征：一般最小-最大表示适用于所有有限单调估值，对于整数加法估值，阈值$T$是剩余自可行的当且仅当每个子集割满足打包-覆盖条件；由于RMMS是剩余自可行份额中的逐点最大值，这些精确常数为基于份额的单独分割算法可实现的公平保证设定了严格限制。
英文摘要：
Residual maximin share (RMMS) is the largest share threshold that remains guaranteeable throughout dynamic allocation processes, even after previously allocated, lower-valued bundles are removed from the item pool. For additive valuations, recent density-balance analyses established finite-agent lower bounds comparing RMMS with the classical maximin share (MMS). In this paper, we prove that these finite-agent lower bounds are exact. Specifically, if $d_n$ denotes the largest odd integer at most $n$, the worst-case ratio satisfies $\inf_{M,v:\operatorname{MMS}>0}\frac{\operatorname{RMMS}(M,v,n)}{\operatorname{MMS}(M,v,n)}=\frac{2d_n}{3d_n-1}$. Consequently, the exact additive frontier forms consecutive odd-even plateaus and converges monotonically to $2/3$. We then investigate the combinatorial structure of extremal instances. While naive witnesses require $Θ(n^2)$ items, we construct an explicit three-valued family achieving the exact boundary with only linear support: $(5n-3)/2$ items for odd $n$ and $(5n-4)/2$ items for even $n$. Its low-valued block supports two exact partitions that simultaneously certify the MMS benchmark and the residual obstruction. By modeling these dual partitions as a bipartite transportation graph, we prove that this block attains the absolute minimum support $q+d-1=3q$. At minimum support, any two-valued filler is uniquely rigid up to relabeling. Finally, we establish structural characterizations of RMMS. A general min--max representation applies to all finite monotone valuations. For integer additive valuations, we prove that a threshold $T$ is residual self-feasible if and only if every subset cut satisfies a packing-covering condition. Because RMMS is pointwise maximal among residual self-feasible shares, these exact constants establish a tight limitation on the fairness guarantees achievable by share-based lone-divider algorithms.
