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

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 $\frac{11}{10} - \varepsilon \approx 1.10$.

• The paper establishes improved NP-hardness of approximation factors for Nash social welfare, budgeted allocation, and GAP via a novel dictator test under the Unique Games Conjecture.
• It introduces a tailored long code-based test that tightens computational gaps, bringing the hardness thresholds closer to known integrality gaps for Configuration LP relaxations.
• These results set strong benchmarks for resource allocation algorithms, indicating that further improvements require innovative techniques beyond traditional PCP and LP approaches.

Improved Hardness of Indivisible Resource Allocation via Unique Games

Overview and Context

This work establishes tighter NP-hardness of approximation results for three critical resource allocation problems with additive valuations:

1. Nash Social Welfare maximization (NSW),
2. Budgeted Allocation, and
3. Generalized Assignment Problem (GAP).

These allocation objectives are central in theoretical computer science and operations research, particularly in resource allocation and algorithmic economics. Prior work produced lower bounds based primarily on Max-E3-Lin-2 reductions or PCP methods, but there remained notable gaps between algorithmic upper bounds, integrality gaps, and hardness thresholds. This paper leverages a novel dictator test and modern reductions under the Unique Games Conjecture (UGC) to close these gaps, providing the strongest known inapproximability factors for the studied problems.

Main Contributions and Technical Results

The core technical innovation is a long code-based dictator test tailored for allocation problems of indivisible goods with additive preferences. By embedding the dictator test within a unique games reduction, the paper delivers the following UGC-based inapproximability results:

• Max Nash Social Welfare (NSW):
  • It is NP-hard to approximate within a factor better than $\sqrt[3]{81/65} - \varepsilon \approx 1.0761$, improving on the prior best $\sqrt{8/7} - \varepsilon \approx 1.069$.
• Max Budgeted Allocation:
  • NP-hard to approximate within $\frac{243}{227} - \varepsilon \approx 1.070$, surpassing the previous best of $\frac{16}{15} - \varepsilon \approx 1.067$.
• Max GAP (Generalized Assignment):
  • NP-hardness of approximation within $\frac{145}{129} - \varepsilon \approx 1.124$, surpassing the prior $\frac{11}{10} - \varepsilon \approx 1.1$.

These are predicated on the UGC and assume additive valuation functions.

Proof Techniques

Novel Dictator Test Construction

The dictator test is key to the paper’s reductions. It constructs instances where allocations correspond to boolean functions, and functions that are dictators (depend only on one input coordinate) can achieve certain “completeness” properties, while functions far from being a dictator are provably limited, via a pairwise independent and balanced distribution over the goods. The construction uses the domain $\{0,1,2\}^R$ to maximize soundness-completeness gap and exploits noise addition for technical applicability of Gaussian stability theorems [Mossel et al.].

Agents and goods are parameterized so that the allocation problem naturally induces a test for presence of dictator functions via allocation-implied labelings, with completeness and soundness corresponding to dictator and non-dictator behaviors, respectively. The hardness is then amplified through a unique games reduction mapping labels to allocation choices.

Meta-Theorem and Application to Objectives

A meta-theorem is developed, characterizing the gap between YES (completeness) and NO (soundness) cases in terms of welfare achievable by non-large-good-receiving agents. Applications to the three objectives involve adjusting the construction (e.g., large good values, group structures, and budget/capacity constraints) to ensure that in the soundness case, even optimal allocations cannot overcome the dictator test’s binding utility cap for non-dictators.

Improvement over Prior Work

Prior hardness bounds generally relied on Max-E3-Lin-2 gadgets and PCP-based reductions, which did not fully leverage modern analytical or test-based Long Code techniques. The present work achieves strictly better inapproximability factors and demonstrates that the hardness closely approaches known integrality gaps for Configuration LP relaxations. For example, for NSW, the configuration LP gap is $2^{1/4} \approx 1.189$, whereas the paper’s result narrows the known computational barrier to within $1.076$.

Key Numerical Hardness Results

Problem	Approx. Hardness (This Work)	Previous Best	Integrality Gap (LP)
Nash Social Welfare	$\approx 1.076$	$\sqrt{8/7} - \varepsilon \approx 1.069$0	$\sqrt{8/7} - \varepsilon \approx 1.069$1
Budgeted Allocation	$\sqrt{8/7} - \varepsilon \approx 1.069$2	$\sqrt{8/7} - \varepsilon \approx 1.069$3	$\sqrt{8/7} - \varepsilon \approx 1.069$4
Generalized Assignment	$\sqrt{8/7} - \varepsilon \approx 1.069$5	$\sqrt{8/7} - \varepsilon \approx 1.069$6	$\sqrt{8/7} - \varepsilon \approx 1.069$7

These values reflect the inapproximability factors under UGC—meaning, unless P=NP or UGC is false, no algorithm achieves a better approximation ratio.

Theoretical and Practical Implications

The theoretical implication is that, conditional on UGC, the computational barriers for these classic allocation problems are now tightly sandwiched between approximation algorithms and integrality gaps, with possibility for further tightening only depending on improved Long Code testing, new pairwise independent distributions, or alternative conjectures.

Practically, these results set strong negative benchmarks for mechanism designers and algorithmic agents: unless one leverages structure beyond additive valuations, or resorts to non-worst-case models, pursuing better approximation guarantees is futile. Additionally, the generalized dictator test technique may catalyze further UGC-based hardness results for other allocation problems (e.g., those with more general utility functions or richer constraints), or even for hybrid objective settings in combinatorial optimization.

Directions for Future Work

• Generalization of Dictator Tests: Exploring alternative balanced pairwise independent distributions for the underlying dictator test could yield sharper or more general bounds.
• Broader Applicability: The reduction technique may apply to fair allocation problems, scheduling with constraints, or hybrid objectives.
• Limitations of Integrality Gap Approaches: As in some cases the LP integrality gap remains above the new lower bound, investigating LP or SDP hierarchies for these settings could reveal tightness.
• Beyond Additivity: Extensions to submodular, budget-additive, or more general valuation classes would be a natural, though technically challenging, next target.

Conclusion

This work advances the understanding of computational hardness for central resource allocation problems, bringing lower bounds near the limits imposed by known relaxations and algorithms, under the powerful hypothesis of the Unique Games Conjecture. The analytic and conceptual tools introduced—a tailored dictator test invoking advanced probability and Boolean function analysis—open new pathways for understanding the limits of approximation in allocation theory and combinatorial optimization.