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 -- such as envy-freeness up to one item (EF1) and minimax share (MMS) -- to our setting and show that fairness is often incompatible with the efficiency notion of social optimality. We then 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. We complement this by designing an XP algorithm (parameterized by the number of agents) for finding MMS and Pareto optimal solutions on every tree instance, and show that the same algorithm can be modified to find efficient solutions along with EF1, when such solutions exist. The latter crucially relies on an intriguing result that in our setting EF1 and Pareto optimality jointly imply MMS. We conclude by theoretically and experimentally analyzing the price of fairness.

翻译：我们首次研究了在图的顶点上布置配送任务时，如何在多个智能体间公平分配配送任务的问题。我们的目标是在满足某些理想的经济效率标准的前提下，将配送成本（建模为次模函数）公平地分配给一组固定的智能体。我们将已确立的公平性概念——如单物品无嫉妒（EF1）和最小最大份额（MMS）——应用于本设定，并证明公平性常与社会最优性这一效率概念不相容。随后，我们通过利用图结构来刻画那些允许存在公平且社会最优解的情形。我们进一步证明，实现公平性与帕累托最优性的计算是难解的。作为补充，我们设计了一个XP算法（以智能体数量为参数），用于在每个树结构实例上寻找MMS且帕累托最优的解，并表明当此类解存在时，该算法可被修改以同时找到满足EF1的有效解。后者关键依赖于一个有趣的结果：在我们的设定中，EF1与帕累托最优性共同蕴含MMS。最后，我们从理论和实验上分析了公平性的代价。