🤖 AI Summary
This paper studies fair cost sharing among multiple agents in graph-structured delivery tasks, where delivery costs are modeled as submodular functions. Addressing the EF1 (envy-freeness up to one item) and MMS (maximin share) fairness criteria—formally introduced here for the first time in graph-structured settings—we characterize graph classes under which EF1 or MMS is compatible with Pareto optimality. We design an XP algorithm for computing MMS+Pareto-optimal allocations on trees and provide a polynomial-time constructive algorithm for EF1-feasible solutions. We prove that fairness and social optimality are generally incompatible, quantifying this trade-off via the Price of Fairness. Experiments validate the efficacy of our algorithms. The core contribution lies in the deep integration of fair allocation theory with graph-structured submodular optimization, unifying theoretical characterization, efficient algorithm design, and empirical evaluation.
📝 Abstract
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. 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 tree instance, and show that the same algorithm can be modified to find efficient solutions along with EF1, when such solutions exist. We complement these results by theoretically and experimentally analyzing the price of fairness.
