We study the problem of allocating indivisible goods under constraints, expressed via a conflict graph $G$. In such an instance, the $m$ items are the vertices of $G$ and connected items cannot be allocated in the same bundle. Under this model, it is already known that EF1 allocations may not exist. Our main contribution is an analysis parametrized by the maximum degree $Δ(G)=Δ$ on the existence and computation of complete EF1 allocations. We address this question in various cases by leveraging results from matching theory. First, we provide a tight existence result for agents with ordered valuations and for the broader class of tiered valuations. We present an algorithm that returns an EF1 allocation when then number of items does not exceed a specific bound. This bound is determined by $n$ and $Δ$, and it is tight when $Δ$ is greater than $2n/3$. We also construct an approximation algorithm when $m$ exceeds this bound. For general additive valuations the problem becomes more challenging. Given the current impossibility results, we focus on the case where the number of items is at most $2n$. For this case, we provide an almost complete picture for the instances that admit EF1 allocations, by combining Round Robin with matchings.

翻译：我们研究了在冲突图 $G$ 约束下的不可分物品分配问题。在此类问题中，$m$ 件物品构成图 $G$ 的顶点，且相连的物品无法被分配到同一束中。在该模型下，已知 EF1 分配可能不存在。我们的主要贡献是分析了最大度 $Δ(G)=Δ$ 参数化下完全 EF1 分配的存在性与计算问题。通过利用匹配理论的结果，我们在多种情形下探讨了该问题。首先，针对具有有序估值以及更广泛的层级估值的智能体，我们给出了紧的存在性结果。当物品数量不超过特定界限时，我们提出了一种返回 EF1 分配的算法。该界限由 $n$ 和 $Δ$ 决定，且当 $Δ$ 大于 $2n/3$ 时是紧的。当 $m$ 超过此界限时，我们还构造了一个近似算法。对于一般的可加估值，问题变得更具挑战性。鉴于当前的不可能性结果，我们重点关注物品数量不超过 $2n$ 的情形。对于该情形，通过将轮询算法与匹配相结合，我们为允许 EF1 分配的实例提供了几乎完整的图景。