A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

TL;DR

This paper presents a strongly polynomial-time algorithm for fair allocation of indivisible items that guarantees Pareto optimality and PROP1 fairness, even with mixed utilities and asymmetric agent weights.

Contribution

It introduces a novel polynomial-time algorithm that ensures Pareto optimality and PROP1 fairness in complex utility settings, resolving an open problem.

Findings

Algorithm guarantees Pareto optimality and PROP1 in polynomial time

Works with mixed utilities and asymmetric agent weights

Does not extend to stronger fairness or efficiency concepts

Abstract

We consider fair allocation of indivisible items under additive utilities. When the utilities can be negative, the existence and complexity of an allocation that satisfies Pareto optimality and proportionality up to one item (PROP1) is an open problem. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. We point out that the result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.