The concept of fair orientations in graphs was introduced by Christodoulou, Fiat, Koutsoupias, and Sgouritsa in 2023, naturally modeling fair division scenarios in which resources are only contested by neighbors. In this model, vertices represent agents and undirected edges represent goods; edges have to be oriented towards one of their endpoints, i.e., allocated to one of their adjacent agents. Although EFX orientations (envy-free up to any good) have been extensively studied in this setting, EF orientations (envy-free) remain unexplored. In this work, we initiate their study, mostly under the lens of parameterized complexity, presenting various tractable cases, hardness results, and parameterizations. Our results concern both simple graphs and multigraphs. Interestingly, many of our results transfer to EFX orientations, thus complementing and improving upon previous work; notably, we answer an open question regarding the structural parameterized complexity of the latter problem on graphs of polynomially-bounded valuations. We also show that EF orientations are tractable in cases in which EFX orientations are not, particularly for binary valuations. Lastly, we consider charity in the orientation setting, establishing algorithms for finding the minimum amount of edges that have to be removed from a graph in order for EF(X) orientations to exist.

翻译：图论中的公平定向概念由 Christodoulou、Fiat、Koutsoupias 和 Sgouritsa 于 2023 年提出，自然地模拟了资源仅被相邻个体争夺的公平分配场景。在此模型中，顶点代表智能体，无向边代表物品；边必须定向到其某一端点，即分配给其相邻的某个智能体。尽管 EFX 定向（对任意物品无嫉妒）在此设定下已被广泛研究，EF 定向（无嫉妒）仍未得到探索。在本工作中，我们首次对其展开研究，主要从参数化复杂性视角出发，提出了多种可处理情形、困难性结果及参数化方案。我们的结果涉及简单图与多重图。有趣的是，许多结果可迁移至 EFX 定向问题，从而补充并改进了先前工作；特别地，我们回答了关于后者在多项式有界估值图上的结构参数化复杂性的开放问题。我们还证明，在 EFX 定向不可处理的情形下（尤其是二值估值情形），EF 定向是可处理的。最后，我们探讨了定向设定中的慈善问题，建立了寻找使 EF(X) 定向存在所需移除的最小边数的算法。