We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.

翻译：我们研究具有冲突约束的不可分割物品的公平分配问题。在此框架下，物品被表示为图的顶点，边对应于物品对之间的冲突关系。每个智能体从图中分配一个独立集。我们的目标是实现这些物品的公平且高效的分配。公平性涉及满足至多一个物品的无嫉妒性（EF1），而高效性由最大化性定义，即没有未分配物品可以可行地分配给任何智能体。首先，我们探索两个智能体的情况。对于单调估值，我们证明在任何图上始终存在一个最大化的EF1分配。存在性证明依赖于一种颜色切换技术，该技术局部修改最大化分配，同时保持可行性并恢复EF1。我们进一步证明，此类分配通常可以在伪多项式时间内计算，而对于任意图上的可加估值以及区间图和二分图上的单调估值，则可在多项式时间内计算。相比之下，一旦放弃单调性，即使对于相同的可加估值，最大化EF1分配也可能不存在，且判断存在性变为NP难问题。接下来，我们考虑一般智能体数量的情况。再次得出负面结论：即使对于三个智能体且具有相同的单调估值，EF1和最大化分配也不存在，且确定此类分配的存在性是NP难的。在正面结果方面，我们证明在路径图上，对于相同的非单调可加估值，始终存在一个EF[1,1]和最大化分配。该结果涉及对“环加三角形”定理的新颖应用。