AI 中文总结
本文研究精确MMS分配存在性的计算复杂性，证明加性估值下为D^P难，2-加性估值下为Δ_2^P完备，并扩展到杂务设置，填补了长期开放问题。
AI 中文摘要
最大最小份额（MMS）保证是分配不可分割物品的一个核心公平性基准。自Kurokawa、Procaccia和Wang [EC'14, JACM'18]证明精确MMS分配不一定存在以来，大量工作研究了近似MMS分配的存在性和计算问题。相比之下，Bouveret和Lemaître [JAAMAS'16]在十多年前提出的一个基本复杂性问答题仍未解决：判定精确MMS分配是否存在有多难？对于加性估值，Lonc和Truszczynski [JAIR'20]证明了该问题属于$\Delta_2^P$（也称为$P^{NP}$），但此前未知任何难度结果。对于更一般的2-加性估值类，Bouveret和Lemaître确立了NP难度，与$\Delta_2^P$上界之间存在显著差距。此外，加性和$k$-加性设置中MMS存在性的（精确）复杂性被列为开放问题。我们在所有这些方面取得了进展：（1）对于加性商品，我们证明了判定MMS存在性是$D^P$-难的，给出了这个长期未解问题的首个难度结果。（2）对于2-加性估值，我们通过证明在单调子模商品实例类上的$\Delta_2^P$-完备性，填补了复杂性差距。据我们所知，这是此类结果的首次出现。我们还证明了三个智能体情况下的弱coNP难度，从而与两个智能体的已知存在性保证建立了精确的二分法；当智能体数量不受限制时，证明了强coNP难度。此外，强难度构造在最优MMS近似比中产生了逆多项式间隙，排除了近似该比率的FPTAS（除非P=NP）。最后，我们展示了所有这些关于商品的结果通过一个保持MMS存在性的多项式时间变换扩展到杂务（chores）设置。
英文摘要
The maximin share (MMS) guarantee has become one of the central fairness benchmarks for allocating indivisible items. Since Kurokawa, Procaccia and Wang~[EC'14, JACM'18] showed that exact MMS allocations need not exist, a substantial literature has developed around the existence and computation of approximate MMS allocations, including the recent work of Heidari, Kaviani, Seddighin, and Shahrezaei [SODA'26]. In contrast, a basic complexity question posed more than a decade ago by Bouveret and Lemaître [JAAMAS'16] has remained unresolved: how hard is it to decide whether an exact MMS allocation exists? For additive valuations, Lonc and Truszczynski [JAIR'20] showed that the problem belongs to the class $Δ_2^P$ (also known as $P^{NP}$), but no hardness result was known. For the more general class of 2-additive valuations, Bouveret and Lemaître established NP-hardness, leaving a substantial gap to the $Δ_2^P$ upper bound. Moreover, the complexity of MMS existence for additive and $k$-additive valuations was posed as open questions. For additive goods, we prove that deciding the existence of an MMS allocation is $Δ_2^P$-complete, settling the complexity of this longstanding problem. We also prove weak coNP-hardness for three agents, thereby establishing a precise dichotomy with the known existence guarantee for two agents, and strong coNP-hardness when the number of agents is unrestricted. Moreover, the strong hardness construction produces an inverse-polynomial gap in the optimal MMS approximation ratio, ruling out an FPTAS for approximating this ratio unless P=NP. We conclude by showing that all these results for goods extend to the chores setting through a polynomial-time transformation that preserves MMS existence.
Comments89 pages; this version strengthens the main result for additive valuations; abstract shortened to meet arXiv requirements
