Improved Hardness Results for Nash Social Welfare, Budgeted Allocation and GAP via the Unique Games Conjecture

2026-05-26

AI Summary

This study addresses the approximability limits of three fundamental problems in fair allocation of indivisible goods under additive valuations: Nash social welfare maximization, budgeted allocation, and the generalized assignment problem (GAP). Building upon the Unique Games Conjecture, the work introduces a novel dictator test tailored to settings of indivisible resource allocation, effectively distinguishing dictator functions from those far from dictators. Leveraging this test, the paper establishes significantly stronger inapproximability lower bounds: Nash social welfare cannot be approximated within a factor better than 1.0761, budgeted allocation within 1.07, and GAP within 1.124. These results markedly improve upon prior hardness guarantees and extend the frontier of complexity theory in combinatorial optimization.

Abstract

We consider the problem of dividing a set of indivisible goods among agents with additive valuations. This problem has been studied under various objectives in both the computer science and the operations research literature. Our main contribution is a novel dictator test using this problem, which can separate a dictator from any function sufficiently far from a dictator. We use this test to prove the following hardness results (assuming the unique games conjecture is true): (1) We show that it is NP-hard to approximate the max Nash welfare by a factor better than $\sqrt[3]{\frac{81}{65}} - \varepsilon \approx 1.0761$. This improves on the previous best known inapproximability factor of $\sqrt{\frac87} - \varepsilon \approx 1.069$. (2) We show that it is NP-hard to approximate the maximum budgeted allocation by a factor better than $\frac{243}{227} - \varepsilon \approx 1.07$. This improves on the previous best known inapproximability factor of $\frac{16}{15} - \varepsilon \approx 1.067$. (3) We show that it is NP-hard to approximate the max generalized assignment problem (GAP) by a factor better than $\frac{145}{129} - \varepsilon \approx 1.124$. This improves on the previous best known inapproximability factor of $\frac{11}{10} - \varepsilon \approx 1.10$.

Problem

• Nash Social Welfare
• Budgeted Allocation
• Generalized Assignment Problem
• Inapproximability
• Unique Games Conjecture

Innovation

• Dictator Test
• Unique Games Conjecture
• Inapproximability
• Nash Social Welfare
• Generalized Assignment Problem