We initiate the study of fair distribution of delivery tasks among a set of agents wherein delivery jobs are placed along the vertices of a graph. Our goal is to fairly distribute delivery costs (modeled as a submodular function) among a fixed set of agents while satisfying some desirable notions of economic efficiency. We adopt well-established fairness concepts$\unicode{x2014}$such as envy-freeness up to one item (EF1) and minimax share (MMS)$\unicode{x2014}$to our setting and show that fairness is often incompatible with the efficiency notion of social optimality. Yet, we characterize instances that admit fair and socially optimal solutions by exploiting graph structures. We further show that achieving fairness along with Pareto optimality is computationally intractable. Nonetheless, we design an XP algorithm (parameterized by the number of agents) for finding MMS and Pareto optimal solutions on every instance, and show that the same algorithm can be modified to find efficient solutions along with EF1, when such solutions exist. We complement our theoretical results by experimentally analyzing the price of fairness on randomly generated graph structures.

翻译：我们开创性地研究了在图中顶点放置配送任务时，如何在一组智能体间公平分配任务的问题。目标是在满足理想经济效益概念的同时，公平分配配送成本（建模为子模函数）。我们将成熟公平性概念（如单物品无嫉妒（EF1）和极小化最大份额（MMS））应用于本场景，并证明公平性与社会最优性效率概念往往不可兼得。然而，我们通过利用图结构刻画了能够同时实现公平与社会最优的实例。进一步研究表明，同时实现公平与帕累托最优在计算上难以处理。尽管如此，我们设计了一个（以智能体数量为参数化）XP算法，用于在任意实例中寻找满足MMS与帕累托最优的解，并证明当公平解存在时，该算法可修改为同时寻找满足EF1与效率的解。我们通过随机图结构上的公平代价实验分析，补充了理论结果。