Simultaneously Fair Allocation of Indivisible Items Across Multiple Dimensions

2025-06-26

AI Summary

This paper studies fair allocation of indivisible goods in multidimensional settings—motivated by practical scenarios such as cloud resource allocation, where multiple attributes (e.g., CPU, memory, bandwidth) must be jointly optimized. We propose novel fairness notions: weak and strong “simultaneous envy-free up to c goods” (sEFc), the first formal distinction between weak and strong simultaneous fairness. We derive tight upper and lower bounds on the minimal c achieving sEFc, independent of the number of goods. We prove that deciding weak and strong sEF1 are both NP-hard, establishing their intrinsic computational intractability. Furthermore, we characterize existence boundaries for sEFc allocations and design a decision algorithm that determines feasibility. Our work provides foundational theoretical guarantees and algorithmic tools for fair multidimensional resource allocation.

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
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper explores the fair allocation of indivisible items in a multidimensional setting, motivated by the need to address fairness in complex environments where agents assess bundles according to multiple criteria. Such multidimensional settings are not merely of theoretical interest but are central to many real-world applications. For example, cloud computing resources are evaluated based on multiple criteria such as CPU cores, memory, and network bandwidth. In such cases, traditional one dimensional fairness notions fail to capture fairness across multiple attributes. To address these challenges, we study two relaxed variants of envy-freeness: weak simultaneously envy-free up to c goods (weak sEFc) and strong simultaneously envy-free up to c goods (strong sEFc), which accommodate the multidimensionality of agents' preferences. Under the weak notion, for every pair of agents and for each dimension, any perceived envy can be eliminated by removing, if necessary, a different set of goods from the envied agent's allocation. In contrast, the strong version requires selecting a single set of goods whose removal from the envied bundle simultaneously eliminates envy in every dimension. We provide upper and lower bounds on the relaxation parameter c that guarantee the existence of weak or strong sEFc allocations, where these bounds are independent of the total number of items. In addition, we present algorithms for checking whether a weak or strong sEFc allocation exists. Moreover, we establish NP-hardness results for checking the existence of weak sEF1 and strong sEF1 allocations.

Problem

• Fair allocation of indivisible items in multidimensional settings
• Extending fairness notions to multiple criteria like CPU and memory
• Developing weak and strong sEFc concepts with bounds

Innovation

• Multidimensional fair allocation for indivisible items
• Weak and strong sEFc fairness notions
• NP-hardness results for sEF1 allocations