We study the fair division problem and the existence of allocations satisfying the fairness criterion envy-freeness up to any item (EFX). The existence of EFX allocations is a major open problem in the fair division literature. We consider binary valuations where the marginal gain of the value by receiving an extra item is either $0$ or $1$. Babaioff et al. [2021] proved that EFX allocations always exist for binary and submodular valuations. In this paper, by using completely different techniques, we extend this existence result to general binary valuations that are not necessarily submodular, and we present a polynomial time algorithm for computing an EFX allocation.

翻译：我们研究了公平分配问题以及满足无嫉妒到任意一个物品（EFX）这一公平准则的分配的存在性。EFX分配的存在性是公平分配领域中的一个重大未解决问题。本文考虑二元估值情形，其中每额外获得一个物品所带来的边际收益为 $0$ 或 $1$。Babaioff等人[2021]证明了对于二元且子模的估值，EFX分配总是存在的。本文采用完全不同的技术手段，将该存在性结果推广至一般二元估值（不必满足子模性），并给出了一个多项式时间算法用于计算EFX分配。