Efficient Nearly-Fair Division with Capacity Constraints

TL;DR

This paper presents a polynomial-time algorithm for fair and efficient allocation of indivisible items with capacity constraints, achieving Pareto-optimality and envy-freeness for two agents under various item valuation scenarios.

Contribution

It introduces a novel polynomial-time algorithm for nearly-fair division with capacity constraints, applicable to two agents and different item valuation types.

Findings

Algorithm finds feasible allocations for two agents with additive utilities.

Allocations are Pareto-optimal and envy-free up to one item in special cases.

The method extends to general cases with mixed goods and chores.

Abstract

We consider the problem of fairly and efficiently allocating indivisible items (goods or bads) under capacity constraints. In this setting, we are given a set of categorized items. Each category has a capacity constraint (the same for all agents), that is an upper bound on the number of items an agent can receive from each category. Our main result is a polynomial-time algorithm that solves the problem for two agents with additive utilities over the items. When each category contains items that are all goods (positively evaluated) or all chores (negatively evaluated) for each of the agents, our algorithm finds a feasible allocation of the items, which is both Pareto-optimal and envy-free up to one item. In the general case, when each item can be a good or a chore arbitrarily, our algorithm finds an allocation that is Pareto-optimal and envy-free up to one good and one chore.