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：我们启动了关于在一组智能体之间公平分配配送任务的研究，其中配送任务沿着图的顶点分布。我们的目标是在满足一些理想的经济效率概念的同时，公平地将配送成本（建模为子模函数）分配给一组固定的智能体。我们将成熟的公平性概念——例如到一项商品为止的无嫉妒性（EF1）和最小化最大份额（MMS）——引入我们的设定中，并表明公平性往往与社会最优性的效率概念不相容。尽管如此，我们通过利用图结构刻画了能够实现公平且社会最优解的实例。我们进一步表明，在实现公平性的同时兼顾帕累托最优性在计算上是棘手的。然而，我们设计了一个XP算法（以智能体数量为参数），用于在每个实例上寻找MMS和帕累托最优解，并证明当存在此类解时，同一算法可以修改以寻找兼顾EF1的高效解。我们通过在随机生成的图结构上实验性分析公平代价来补充我们的理论结果。