Fair Allocation of Service Tasks with Two-Dimensional Costs

Hadi Hosseini, Minming Li, Shivika Narang, Zhehan Yu

Keywords: Fair Allocations, EF1, MMS

TL;DR: We study the existence of fair and efficient allocations of tasks on a graph when costs come from both edges and vertices.

Abstract: Recent work has explored fair allocations of delivery tasks where delivery agents incur costs based on the distance they travel. We generalize this setting to one of service tasks where an agent's cost has two dimensions: the time spent completing each task and the distance traveled. Thus, the input includes a graph where all nodes other than the hub correspond to a unique order/task, and must be allocated to some agent. We model the cost incurred by an agent as a linear function of the edges traversed and the nodes they need to service. In this setting, we explore two well motivated fairness concepts: Envy-Freeness up to One Order (EF1) and Minimax Share (MMS). We show several surprising results, including the fact that in our setting an MMS allocation can be found in polynomial time. We also show conditions which can guarantee the existence of allocations that are both EF1 and MMS. We further show that these allocations can be found in polynomial time. We also provide tight upper bounds on the price of fairness. We complement our theoretical results with an experimental analysis demonstrating the effect of various input parameters on the MMS cost.

Submission Number: 8