We consider the problem of sharing a set of indivisible goods among agents in a fair manner, namely such that the allocation is envy-free up to any good (EFX). We focus on the problem of computing an EFX allocation in the two-agent case and characterize the computational complexity of the problem for most well-known valuation classes. We present a simple greedy algorithm that solves the problem when the agent valuations are weakly well-layered, a class which contains gross substitutes and budget-additive valuations. For the next largest valuation class we prove a negative result: the problem is PLS-complete for submodular valuations. All of our results also hold for the setting where there are many agents with identical valuations.

翻译：我们研究在多个代理之间公平共享一组不可分割物品的问题，即分配需达到直到任意物品无嫉妒性（EFX）。我们聚焦于双代理情形下计算EFX分配的问题，并刻画了大多数经典估值类别的计算复杂性。我们提出一种简单贪心算法，可在代理估值属于弱良分层类（包含总替代性和预算可加性估值）时求解该问题。针对次大估值类别，我们证明了否定性结果：对于次模估值，该问题属于PLS完全类。我们的所有结论同样适用于存在多个具有相同估值的代理的场景。