公平且高效的分配：多项式层级间隙中的决策问题
Fair and Efficient Allocations: Decision Problems in the Gap of Polynomial Hierarchy
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中文总结 AI 辅助
本文研究公平分配中无嫉妒且高效的分配存在性决策问题，刻画了其计算复杂性，发现许多版本落入多项式层级间隙，并扩展了已有结果至更受限设置。
中文摘要 AI 辅助
我们考虑具有不可分割商品的公平分配问题，并研究以下决策问题：给定一个公平分配实例，是否存在一个既无嫉妒又高效的分配？我们考虑两种效率标准：帕累托最优性和社会福利最优性。我们提供了该决策问题计算复杂性的完整图景，其中代理人数从$2$到$\infty$不等，既包括加性估值，也包括一般估值，以及更受限的$k$元估值函数类别（其中物品的边际值被限制为$\{0,1,\ldots,k-1\}$，对于某个常数$k\geq2$）。一个有趣的观察是，上述决策问题的许多版本落入多项式层级第一级和第二级之间的“间隙”中。具体来说，假设多项式层级不坍缩到第一级（即假设$\text{NP}\neq\text{coNP}$），这些问题属于$(\Sigma_2^{\text{p}}\cap\Pi_2^{\text{p}})\setminus(\text{NP}\cup\text{coNP})$。特别是，我们提供了跨不同参数范围（包括代理人数和估值模型的选择）的细粒度复杂性分析。根据不同的参数，许多问题具有不同的复杂性分类，从介于两级之间的中间类$\Theta_2^{\text{p}}$和$\Delta_2^{\text{p}}$到$\Sigma_2^{\text{p}}$-完全性。最后，De Keijzer等人证明了在加性估值下以帕累托最优性作为效率标准的决策问题是$\Sigma_2^{\text{p}}$-完全的。我们的主要结果将此扩展至更受限的设置，例如具有常数个代理人的实例或$3$元估值函数，这解决了Bouveret和Lang提出的开放问题。
英文摘要
We consider the fair division problem with indivisible goods and study the following decision problem: given a fair division instance, does there exist an allocation that is envy-free and efficient? We consider two efficiency criteria: Pareto-optimality and social welfare optimality. We provide a complete landscape on the computational complexity of this decision problem, with the number of agents ranging from $2$ to $\infty$, both additive valuations and general valuations, and the more restricted class of $k$-ary valuation functions (where an item's marginal value is restricted to $\{0,1,\ldots,k-1\}$ for some constant $k\geq2$). One interesting observation is that many versions of the above-mentioned decision problems fall into the ``gap'' between the first and the second levels of the polynomial hierarchy. Specifically, assuming the polynomial hierarchy does not collapse to the first level (i.e., assuming $\text{NP}\neq\text{coNP}$), these problems are in $(Σ_2^{\text{p}}\capΠ_2^{\text{p}})\setminus(\text{NP}\cup\text{coNP})$. In particular, we provide a fine-grained complexity analysis across different parameter regimes, including the number of agents and the choice of valuation models. Depending on different parameters, many problems admit different complexity classifications, ranging from the intermediate classes $Θ_2^{\text{p}}$ and $Δ_2^{\text{p}}$ between the two levels to $Σ_2^{\text{p}}$-completeness. Finally, De Keijzer et al. show the $Σ_2^{\text{p}}$-completeness of the decision problem when considering Pareto-optimality as the efficiency criterion with additive valuations. Our main results extend this result to more restricted settings, such as instances with a constant number of agents or $3$-ary valuation functions, which resolves the open problem given by Bouveret and Lang.
发表机构
- Shanghai Jiao Tong University（上海交通大学）
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