We provide a new mathematical framework for the classic problem of fair allocation of indivisible goods, showing that it can be formulated as the problem of finding an optimal column rearrangement of multiple matrices. Based on this formulation, we design two novel algorithms called MinCov and MinCovTarget to find optimal allocations under the newly introduced notion of minimum social inequality, and the popular notion of minimum envy. Numerical illustrations show an excellent performance of the newly developed algorithms also with respect to other allocation criteria, in particular for the maximum Nash welfare.

中文翻译：

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以最小的不平等或最小的嫉妒公平分配不可分割的商品

我们为不可分割物品公平分配的经典问题提供了一个新的数学框架，表明它可以被表述为寻找多个矩阵的最佳列重新排列的问题。基于此公式，我们设计了两种名为 MinCov 和 MinCovTarget 的新算法，以在新引入的最小社会不平等概念和流行的最小嫉妒概念下找到最佳分配。数值插图显示了新开发的算法在其他分配标准方面的出色表现，尤其是在最大纳什福利方面。