发表机构
The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对不可分割商品分配中的无嫉妒问题，提出了基于匹配和最短路径的立方时间算法，并给出了最小补贴的完整渐近刻画，覆盖不同余数情形。
AI 中文摘要
我们研究了在将 $m_n$ 个不可分割商品分配给 $n$ 个智能体时，实现精确无嫉妒所需的最小货币补贴。估值是可加的，商品价值独立同分布，取自 $[0,1]$ 上的一个分布，其密度有上界且远离零，所有智能体对金钱的估值相同。我们考虑 $n\to\infty$ 且 $m_n=qn+r_n$ 的情形，其中 $q\ge0$ 是固定整数，$0\le r_n<n$。对于非零余数情形（即 $1\le r_n<n$），我们给出一个基于平衡束的最大权重匹配和互补对偶价格的三次时间算法。该算法对每个实例都返回一个无嫉妒的结果。在随机模型下，当 $q\ge1$ 时，其支付额和无限制最小值均为 $n-r_n+o_p(n)$。当 $q=0$ 且 $r_n/n$ 收敛到小于 1 的极限时，误差项改进为 $O_p(1)$。对于精确可分情形（即 $r_n=0$），先前的工作表明当 $q\ge2$ 时，高概率下零补贴即可实现。平方情形 $q=1$ 是特殊的。其最小补贴为 $\Theta_p(\log n)$，并且高概率下等于在所有智能体恰好获得一个商品的所有分配方案中的最优值。这个受限最优值可以通过最大权重完美匹配和全对最短路径在三次时间内计算。综合来看，这些结果为最小补贴提供了统一的渐近刻画。
英文摘要
We study the minimum monetary subsidy required for exact envy-freeness in allocating $m_n$ indivisible goods to $n$ agents. Valuations are additive, item values are i.i.d. from a distribution on $[0,1]$ with density bounded above and away from zero, and all agents value money equally. We consider $n\to\infty$ with $m_n=qn+r_n$, where $q\ge0$ is a fixed integer and $0\le r_n<n$. For the nonzero-remainder case where $1\le r_n<n$, we give a cubic-time algorithm based on a maximum-weight matching of balanced bundles and complementary dual prices. It returns an envy-free outcome for every instance. Under the random model, both its payment and the unrestricted minimum are $n-r_n+o_p(n)$ for $q\ge1$. When $q=0$ and $r_n/n$ converges to a limit below one, the error term improves to $O_p(1)$. For exact divisibility, where $r_n=0$, prior work gives zero subsidy with high probability for $q\ge2$. The square case $q=1$ is exceptional. Its minimum subsidy is $Θ_p(\log n)$ and, with high probability, equals the optimum over allocations in which every agent receives exactly one good. This restricted optimum can be computed in cubic time using a maximum-weight perfect matching and all-pairs shortest paths. Together, these results provide a unified asymptotic characterization of the minimum subsidy.
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