发表机构
National Economics University; Khalifa University of Science and Technology（国民经济大学; 哈利法科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究高多重性设置下$p$-均值社会福利最大化，证明少数物品类型时NP-hard，并给出少数类型下的FPTAS和PTAS，基本解决其复杂性与可近似性。
AI 中文摘要
$p$-均值福利目标统一了不可分割物品分配中的若干经典社会福利准则。我们在物品和智能体的多重性以二进制编码时，研究在加性非负效用下的最大化问题。对于每个固定的有限有理数 $p<1$，$p\neq0$，我们证明该问题在仅有两种物品类型时是 $\mathsf{NP}$-困难的。纳什福利（$p=0$）和平均主义福利（$p=-\infty$）在三种物品类型下是 $\mathsf{NP}$-困难的。这些结果既适用于计算最优分配，也适用于有理阈值决策，且效用和阈值可以被要求为正整数。我们还证明，在一种智能体类型和不受限制的物品类型数量下，对于每个固定的有限有理数 $p<1$ 以及 $p=-\infty$，最大化 $p$-均值社会福利是强 $\mathsf{NP}$-困难的。该归约中的定量差距排除了后一种设置中的 FPTAS，除非 $\mathsf{P}=\mathsf{NP}$。在积极方面，对于固定数量的物品类型和任意数量的智能体类型，我们为每个固定的 $p\in\mathbb Q\cup\{-\infty\}$ 给出了一个 FPTAS。其运行时间在紧凑输入长度和 $1/\varepsilon$ 上是多项式的，并返回一个压缩分配。对于固定数量的智能体类型和不受限制的物品类型数量，我们为每个固定的有限有理数 $p<1$ 给出了一个 PTAS，同样在完全紧凑模型中。我们还给出了经典精确分配算法在一种物品类型下以及两种物品类型下平均主义福利的显式紧凑模型证明。这些结果基本上解决了在少数物品类型和/或少数智能体类型下 $p$-均值福利最大化的复杂性和可近似性，仅在小型物品类型分类中留下了两种物品类型下纳什福利最大化的精确复杂性未解决。
英文摘要
The $p$-mean welfare objective unifies several classical social welfare criteria for the allocation of indivisible goods. We study its maximization under additive nonnegative utilities when item and agent multiplicities are encoded in binary. For every fixed finite rational $p<1$, $p\neq0$, we show that the problem is $\mathsf{NP}$-hard with only two item types. Nash welfare ($p=0$) and egalitarian welfare ($p=-\infty$) are NP-hard with three item types. These results hold both for computing an optimal allocation and for rational-threshold decision, and the utilities and threshold can be required to be positive integers. We also show that maximizing $p$-mean social welfare is strongly NP-hard with one agent type and an unrestricted number of item types, for every fixed finite rational $p<1$ and for $p=-\infty$. A quantitative gap in this reduction rules out an FPTAS in the latter setting unless $\mathsf{P}=\mathsf{NP}$. On the positive side, for a fixed number of item types and an arbitrary number of agent types, we give an FPTAS for every fixed $p\in\mathbb Q\cup\{-\infty\}$. Its running time is polynomial in the compact input length and in $1/\varepsilon$, and it returns a compressed allocation. For a fixed number of agent types and an unrestricted number of item types, we give a PTAS for every fixed finite rational $p<1$, also in the fully compact model. We further give explicit compact-model proofs of the classical exact allocation algorithms for one item type, and for egalitarian welfare with two item types. These results essentially settle the complexity and approximability of $p$-mean welfare maximization with few item types and/or few agent types, leaving only the exact complexity of Nash welfare maximization with two item types unresolved in the small-item-type classification.
