We investigate optimal social welfare allocations of $m$ items to $n$ agents with binary additive or submodular valuations. For binary additive valuations, we prove that the set of optimal allocations coincides with the set of so-called \emph{stable allocations}, as long as the employed criterion for evaluating social welfare is strongly Pigou-Dalton (SPD) and symmetric. Many common criteria are SPD and symmetric, such as Nash social welfare, leximax, leximin, Gini index, entropy, and envy sum. We also design efficient algorithms for finding a stable allocation, including an $O(m^2n)$ time algorithm for the case of indivisible items, and an $O(m^2n^5)$ time one for the case of divisible items. The first is faster than the existing algorithms or has a simpler analysis. The latter is the first combinatorial algorithm for that problem. It utilizes a hidden layer partition of items and agents admitted by all stable allocations, and cleverly reduces the case of divisible items to the case of indivisible items. In addition, we show that the profiles of different optimal allocations have a small Chebyshev distance, which is 0 for the case of divisible items under binary additive valuations, and is at most 1 for the case of indivisible items under binary submodular valuations.

翻译：我们研究了具有二元可加或次模估值的n个代理人对m件物品的最优社会福利分配问题。对于二元可加估值，我们证明只要所采用的社会福利评估准则满足强Pigou-Dalton（SPD）对称性，最优分配集合即与所谓的稳定分配集合完全一致。许多常见准则均满足SPD对称性，例如纳什社会福利、字典序最大值、字典序最小值、基尼系数、熵以及嫉妒总和。我们还设计了寻找稳定分配的高效算法，包括针对不可分物品情况的O(m²n)时间算法，以及针对可分物品情况的O(m²n⁵)时间算法。前者较现有算法速度更快或分析更简洁，后者是该问题的首个组合算法。该算法利用所有稳定分配所允许的物品与代理人的隐藏层划分，并巧妙地将可分物品情况归约为不可分物品情况。此外，我们证明不同最优分配的配置具有较小的切比雪夫距离：在二元可加估值下的可分物品情况下该距离为0，在二元次模估值下的不可分物品情况下该距离至多为1。