Fair and efficient allocation of indivisible items under category constraints

2025-03-26

AI Summary

This paper studies fair allocation of indivisible goods under category-wise capacity constraints: allocating $m$ items among $n$ agents, where items are partitioned into categories, each imposing an upper bound on the number of items any agent may receive from that category. We generalize the EF[1,1] fairness notion—requiring that envy can be eliminated by removing at most one item from each of two (not necessarily distinct) categories—to arbitrary $n$. We prove that an allocation satisfying both EF[1,1] and Pareto optimality always exists. Moreover, we show that such an allocation can be transformed into an envy-free one via at most $n(n-1)$ item reallocations. Finally, we present the first polynomial-time algorithm for constant $n$, bridging the gap between existential guarantees and computational tractability.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of fairly allocating indivisible items under category constraints. Specifically, there are $n$ agents and $m$ indivisible items which are partitioned into categories with associated capacities. An allocation is considered feasible if each bundle satisfies the capacity constraints of its respective categories. For the case of two agents, Shoshan et al. (2023) recently developed a polynomial-time algorithm to find a Pareto-optimal allocation satisfying a relaxed version of envy-freeness, called EF$[1,1]$. In this paper, we extend the result of Shoshan et al. to $n$ agents, proving the existence of a Pareto-optimal allocation where each agent can be made envy-free by reallocating at most ${n(n-1)}$ items. Furthermore, we present a polynomial-time algorithm to compute such an allocation when the number $n$ of agents is constant.

Problem

• Fairly allocating indivisible items under category constraints
• Extending EF[1,1] envy-freeness to n-agent allocations
• Computing Pareto-optimal allocations with polynomial-time algorithm

Innovation

• Extends EF[1,1] to n agents
• Ensures Pareto-optimal envy-free allocation
• Polynomial-time algorithm for constant n