We study the fair division of indivisible goods with conflicts between pairs of goods, represented by a graph $G = (V, E)$. We consider ``soft'' conflicts: assigning two adjacent goods to the same agent is allowed, but we seek allocations that are envy-free up to one good (EF1) while keeping the number of such conflict violations small. We propose a linear-time algorithm for general additive valuations that finds an EF1 allocation with at most $|E|/n + O(|E|^{1-1/(2n-2)})$ violations, for any constant number of agents $n$. The leading term $|E|/n$ matches the worst-case bound on the number of violations. We use a novel approach that combines an algorithm for fair division with cardinality constraints from Biswas \& Barman (2018) and a geometric ``closest points'' argument. For identical additive valuations, we also propose a simple round-robin-based algorithm that finds an EF1 allocation with at most $|E|/n$ violations.

翻译：我们研究了不可分割物品的公平分配问题，其中物品之间存在冲突，这些冲突由图$G = (V, E)$表示。我们考虑“软”冲突：允许将两个相邻的物品分配给同一智能体，但我们寻求的分配方案需满足“除一物品外无嫉妒”（EF1）条件，同时使此类冲突违规的数量尽可能少。针对一般加性估值，我们提出了一种线性时间算法，该算法能为任意常数个智能体$n$找到一个EF1分配，其违规数至多为$|E|/n + O(|E|^{1-1/(2n-2)})$。首项$|E|/n$与违规数量的最坏情况界相匹配。我们采用了一种新颖的方法，该方法结合了Biswas \& Barman（2018）提出的具有基数约束的公平分配算法和一个几何上的“最近点”论证。对于相同的加性估值，我们还提出了一种基于简单循环轮转的算法，该算法能找到违规数至多为$|E|/n$的EF1分配。