Envy-free Allocations with Individual Payments

2026-10-07

AI Summary

This study addresses the envy-free fair allocation of indivisible items under individual budget constraints without external subsidies, permitting only restricted monetary transfers among agents. Employing a parameterized complexity analysis framework, this work designs fixed-parameter tractable (FPT) algorithms alongside polynomial-time verification procedures. The results establish that while the problem is NP-hard in its general form, it becomes efficiently solvable for specific parameters by demonstrating parameterized tractability grounded in budget constraints. By departing from the conventional assumption of external subsidies, this research resolves pertinent open theoretical questions and advances the theoretical foundations of constrained fair division.

Abstract

When an envy-free allocation of indivisible goods does not exist, monetary transfers can restore envy-freeness. Existing work on fair division with subsidies, however, typically assumes that these payments are provided by an external source, an assumption that may be unrealistic in many applications. We address this limitation by allowing only monetary transfers between agents, with each agent's payments constrained by their individual budget. We show that while it is polynomial-time tractable to determine whether a given allocation can be made envy-free by payments under individual budgets, the general problem of computing such an allocation from scratch is NP-hard, even when agents have relatively large budgets. Motivated by this intractability, we conduct a parameterized complexity analysis, establishing fixed-parameter tractability with respect to the number of goods or to the joint parameter number of agents and good types, and we provide efficient algorithms in several special cases. For explicitly listed items, our type-based algorithm answers the envy-freeness part of an open question of T. T. Nguyen and J. Rothe, "Complexity Results and Exact Algorithms for Fair Division of Indivisible Items: A Survey."

Problem

• envy-free allocation
• indivisible goods
• individual payments
• fair division
• budget constraint

Innovation

• envy-free allocation
• individual budgets
• parameterized complexity
• fixed-parameter tractability
• monetary transfers