可弃权估值的简单多项式时间EFX修复算法
A Simple Polynomial-Time EFX Repair for Cancelable Valuations
中文总结 AI 辅助
本文提出一种适用于可弃权估值的多项式时间EFX修复算法，将其应用于受限加法负担分配时，可同时满足EFX、(1+ε)-MMS及最优社会成本的2-近似，且性能优于此前的4/3-MMS保证。
中文摘要 AI 辅助
Plaut和Roughgarden针对具有相同单调估值的智能体提出的leximin++证明给出了一种自然的EFX（无嫉妒性的最大最小份额）修复程序：从任意划分开始，反复将合格物品转移到价值最低的束中。该程序会终止，但即使对于加法估值，标准论证也未给出转移次数的多项式界。我们表明，单一确定性平局打破规则可使该修复程序对于更广泛的可弃权（cancelable）估值类成为多项式时间算法。固定与物品单例价值一致的物品排序，且始终转移排名最高的合格物品。连续转移的物品在该排序中严格递减，因此算法最多执行m次转移，其中m为物品数量。此外，该修复程序不会降低最小束价值或增加最大束价值。作为应用，对于每个固定的ε>0，我们在多项式时间内计算出受限加法负担（chores）的分配，该分配同时满足EFX、(1+ε)-MMS（最大最小份额）以及最优社会成本的2-近似，这改进了之前多项式时间的4/3-MMS保证。最后，我们展示了一个包含5个物品的单调可弃权排序，其无加法表示，表明超出加法性的扩展是真实存在的。
英文摘要
The leximin++ proof of Plaut and Roughgarden for agents with identical monotone valuations gives a natural EFX-repair procedure: starting from an arbitrary partition, repeatedly transfer an eligible item to a minimum-valued bundle. The procedure terminates, but the standard argument gives no polynomial bound on the number of transfers, even for additive valuations. We show that a single deterministic tie-breaking rule makes this repair procedure polynomial for the broader class of cancelable valuations. Fix an ordering of the items consistent with their singleton values and always transfer the highest-ranked eligible item. Consecutive transferred items strictly decrease in this ordering, and hence the algorithm performs at most $m$ transfers, where $m$ is the number of items. Moreover, the repair procedure does not decrease the minimum bundle value or increase the maximum bundle value. As an application, for every fixed $\varepsilon>0$, we compute in polynomial time an allocation of restricted additive chores that is simultaneously EFX, $(1+\varepsilon)$-MMS, and a $2$-approximation to the optimal social cost. This improves upon the previous polynomial-time $4/3$-MMS guarantee. Finally, we exhibit a monotone cancelable ordering on five items with no additive representation, showing that the extension beyond additivity is genuine.
