We consider the problem of allocating $m$ indivisible items to a set of $n$ heterogeneous agents, aiming at computing a proportional allocation by introducing subsidy (money). It has been shown by Wu et al. (WINE 2023) that when agents are unweighted a total subsidy of $n/4$ suffices (assuming that each item has value/cost at most $1$ to every agent) to ensure proportionality. When agents have general weights, they proposed an algorithm that guarantees a weighted proportional allocation requiring a total subsidy of $(n-1)/2$, by rounding the fractional bid-and-take algorithm. In this work, we revisit the problem and the fractional bid-and-take algorithm. We show that by formulating the fractional allocation returned by the algorithm as a directed tree connecting the agents and splitting the tree into canonical components, there is a rounding scheme that requires a total subsidy of at most $n/3 - 1/6$.

翻译：我们研究将$m$个不可分割物品分配给$n$个异质智能体的问题，旨在通过引入补贴（货币）实现比例分配。Wu等人（WINE 2023）已证明，当智能体无权重时（假设每个物品对每个智能体的价值/成本至多为1），总额为$n/4$的补贴足以确保比例性。对于智能体具有一般权重的情形，他们通过对分数形式的投标-取用算法进行取整，提出了一种保证加权比例分配且所需补贴总额为$(n-1)/2$的算法。本文重新审视该问题与分数形式的投标-取用算法。我们证明，通过将算法返回的分数分配表述为连接各智能体的有向树，并将该树分割为规范分量，存在一种取整方案，其所需补贴总额不超过$n/3 - 1/6$。