Resource Allocation under the Latin Square Constraint

2025-01-11

AI Summary

This paper studies fair resource allocation under Latin square constraints: allocating $n$ indivisible items to $n$ agents over $n$ rounds, where each agent receives at most one item per round and each item is allocated at most once. The objectives are to maximize either the utilitarian social welfare (sum of utilities) or the egalitarian social welfare (minimum agent utility). It introduces Latin square structures to fair division for the first time, proposing approximation algorithms achieving $(1-1/e)$-approximation for utilitarian welfare and $(1-1/e)/4$-approximation for egalitarian welfare, as well as an FPT algorithm parameterized by both the order $n$ and the optimal welfare value. Furthermore, it establishes NP-completeness for verifying envy-freeness, proportionality, and several other fairness criteria under binary valuations. The approach integrates techniques from combinatorial optimization, approximation algorithms, fixed-parameter tractability analysis, and computational complexity theory.

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

A Latin square is an $n imes n$ matrix filled with $n$ distinct symbols, each of which appears exactly once in each row and exactly once in each column. We introduce a problem of allocating $n$ indivisible items among $n$ agents over $n$ rounds while satisfying the Latin square constraint. This constraint ensures that each agent receives no more than one item per round and receives each item at most once. Each agent has an additive valuation on the item--round pairs. Real-world applications like scheduling, resource management, and experimental design require the Latin square constraint to satisfy fairness or balancedness in allocation. Our goal is to find a partial or complete allocation that maximizes the sum of the agents' valuations (utilitarian social welfare) or the minimum of the agents' valuations (egalitarian social welfare). For the problem of maximizing utilitarian social welfare, we prove NP-hardness even when the valuations are binary additive. We then provide $(1-1/e)$ and $(1-1/e)/4$-approximation algorithms for partial and complete settings, respectively. Additionally, we present fixed-parameter tractable (FPT) algorithms with respect to the order of Latin square and the optimum value for both partial and complete settings. For the problem of maximizing egalitarian social welfare, we establish that deciding whether the optimum value is at most $1$ or at least $2$ is NP-hard for both the partial and complete settings, even when the valuations are binary. Furthermore, we demonstrate that checking the existence of a complete allocation that satisfies each of envy-free, proportional, equitable, envy-free up to any good, proportional up to any good, or equitable up to any good is NP-hard, even when the valuations are identical.

Problem

• Optimal Resource Allocation
• Fair Distribution
• Valuation Maximization

Innovation

• Latin Square Allocation
• Optimal Distribution
• Fairness in Resource Management