Simultaneously Efficient Allocation of Indivisible Items Across Multiple Dimensions

2026-06-19

AI Summary

This work addresses the imbalance in multidimensional efficiency that arises when optimizing a single objective in the allocation of indivisible goods. We propose a Multidimensionally Efficient Allocation (MDEA) model to jointly optimize utilitarian social welfare (USW) and egalitarian social welfare (ESW) under additive valuations across dimensions. We introduce three notions of multidimensional Pareto efficiency, characterize their interrelationships and computational complexity, and analyze them through the lenses of combinatorial optimization, approximation algorithms, and multiobjective utility theory. Our main contributions include establishing a tight $c/\ell$-approximation for USW, showing that the dependence on the number of dimensions $\ell$ is unavoidable; proving that ESW is already NP-hard in two dimensions; and identifying $1/\ell$ as the tight threshold for approximating efficiency across all dimensions.

Abstract

Many allocation problems are intrinsically multidimensional, since an item may contribute differently to several criteria, and optimizing a single aggregate objective can hide severe losses in other dimensions. We study how much efficiency can be guaranteed simultaneously when indivisible items have multiple attributes. To this end, we introduce the \emph{multidimensional efficient allocation} (MDEA) model, where each agent has an additive valuation in each dimension, and investigate simultaneous efficiency under utilitarian social welfare (USW) and egalitarian social welfare (ESW). Our results reveal a sharp worst-case frontier. For exact efficiency, maximizing the number of dimensions attaining the USW optimum admits a $c/\ell$-approximation for every fixed constant $c$, and this dependence on the number $\ell$ of dimensions is essentially unavoidable; for ESW, even deciding whether two dimensions can be optimized simultaneously is NP-hard with binary valuations. For approximate simultaneous efficiency in every dimension, we identify a tight threshold of order $1/\ell$, showing that such guarantees always exist for both USW and ESW, while any asymptotically better dependence on $\ell$ is impossible, even for binary valuations. Finally, we introduce three natural multidimensional Pareto notions and characterize both their relationships and their computational complexity.

Problem

• multidimensional allocation
• indivisible items
• simultaneous efficiency
• social welfare
• Pareto efficiency

Innovation

• multidimensional efficient allocation
• simultaneous efficiency
• indivisible goods
• social welfare
• Pareto optimality