This paper addresses the problem of finding fair orientations of graphs of chores, in which each vertex corresponds to an agent, each edge corresponds to a chore, and a chore has zero marginal utility to an agent if its corresponding edge is not incident to the vertex corresponding to the agent. Recently, Zhou et al. (IJCAI, 2024) analyzed the complexity of deciding whether graphs containing a mixture of goods and chores have EFX orientations, and conjectured that deciding whether graphs containing only chores have EFX orientations is NP-complete. We resolve this conjecture by giving polynomial-time algorithms that find EF1 and EFX orientations of graphs containing only chores if they exist, even if there are self-loops. Remarkably, our result demonstrates a surprising separation between the case of goods and the case of chores, because deciding whether graphs containing only goods have EFX orientations was shown to be NP-complete by Christodoulou et al. (EC, 2023). In addition, we show the EF1 and EFX orientation problems for multigraphs to be NP-complete.

翻译：本文研究家务图的公平定向问题，其中每个顶点对应一个智能体，每条边对应一项家务，且当某条边不与某智能体对应的顶点邻接时，该家务对该智能体的边际效用为零。最近，Zhou等人（IJCAI, 2024）分析了判断包含物品与家务混合的图是否存在EFX定向的复杂度，并猜想判断仅包含家务的图是否存在EFX定向是NP完全问题。我们通过给出多项式时间算法解决了该猜想，该算法能在存在解时（即使存在自环）找到仅包含家务的图的EF1与EFX定向。值得注意的是，我们的结果揭示了物品情形与家务情形之间惊人的分离性，因为Christodoulou等人（EC, 2023）已证明判断仅包含物品的图是否存在EFX定向是NP完全问题。此外，我们证明了多重图的EF1与EFX定向问题均为NP完全问题。