Ensuring fairness while limiting costs, such as transportation or storage, is an important challenge in resource allocation, yet most work has focused on cost minimization without fairness or fairness without explicit cost considerations. We introduce and formally define the minCost-EFx Allocation problem, where the objective is to compute an allocation that is envy-free up to any item (EFx) and has minimum cost. We investigate the algorithmic complexity of this problem, proving that it is NP-hard already with two agents. On the positive side, we show that the problem admits a polynomial kernel with respect to the number of items, implying that a core source of intractability lies in the number of items. Building on this, we identify parameter-restricted settings that are tractable, including cases with bounded valuations and a constant number of agents, or a limited number of item types under restricted cost models. Finally, we turn to cost approximation, proving that for any $ρ>1$ the problem is not $ρ$-approximable in polynomial time (unless $P=NP$), while also identifying restricted cost models where costs are agent-specific and independent of the actual items received, which admit better approximation guarantees.

翻译：在资源分配中，确保公平性同时控制成本（如运输或存储成本）是一项重要挑战，然而现有研究大多集中于不考虑公平性的成本最小化或不考虑显式成本约束的公平性。我们引入并形式化定义了最小成本-EFx分配问题，其目标是计算一个满足任意物品无嫉妒（EFx）且成本最小的分配方案。我们研究了该问题的算法复杂度，证明即使只有两个智能体，该问题已是NP难的。从积极方面看，我们证明了该问题关于物品数量存在多项式核，这意味着计算困难的核心根源在于物品数量。基于此，我们识别出一些可处理的参数限制场景，包括有界估值与常数个智能体的情形，或在受限成本模型下物品类型数量有限的情形。最后，我们转向成本近似性研究，证明对于任意$ρ>1$，该问题不存在多项式时间的$ρ$近似算法（除非$P=NP$），同时也识别出一些受限成本模型（其中成本与智能体相关且与所获具体物品无关），这些模型可获得更好的近似保证。