Ternary Valuations and Fair Allocation Complexity Analysis

Core Concepts

Fair allocation under ternary valuations is APX-hard for both Nash and egalitarian welfare objectives.

Abstract

The content discusses the complexity of fair allocation under ternary valuations with distinct values a, b, and c. It explores the challenges of maximizing Nash and egalitarian welfare, showing APX-hardness for both objectives. The analysis includes reductions from graph theory problems and comparisons with previous results.

Abstract

• Fair allocation of indivisible items with ternary valuations is studied.
• Max Nash welfare (MNW) and max egalitarian welfare (MEW) are considered.

Introduction

• Fair allocation is a fundamental issue in computational economics.
• Challenges arise when agents have arbitrary valuations.

Results

• APX-hardness is shown for maximizing Nash welfare under ternary valuations.
• Complexity analysis for maximizing egalitarian welfare is provided.

Additional Work

• Approximation algorithms and APX-hardness in related studies are discussed.

Preliminaries

• Definitions and assumptions for the fair allocation problem.

The All Goods Case: 0 ≤ a < b < c

• Complexity analysis for MNW allocation under specific valuations.

Further Complexity Analysis

• Resolution of APX-hardness for MNW allocation under different valuation scenarios.

Conclusion

• The impossibility of approximating MNW under specific valuations is highlighted.

Stats

For any distinct a, b, and c, maximizing Nash welfare is APX-hard.
For any distinct a, b, and c, maximizing egalitarian welfare is APX-hard.

Quotes

"Recent works study simpler classes of valuations where exact fair allocations can be computed."
"Fair allocation of indivisible items is a fundamental problem in computational economics."

Key Insights Distilled From

On the Hardness of Fair Allocation under Ternary Valuations

by Zack Fitzsim... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.00943.pdf

Deeper Inquiries

본 연구에서는 공정 배분 문제의 복잡성을 다양한 가치 평가 시나리오에서 더 줄일 수 있을까요?

이 연구에서는 특정한 세 가지 값 a, b, c에 대한 가치평가 시나리오에서 최대 나시 복지(MNW) 문제의 복잡성을 분석했습니다. 그러나 다른 가치 평가 시나리오에서는 더 낮은 복잡성을 가질 수 있습니다. 예를 들어, 다양한 가치 범위, 추가적인 제약 조건 또는 다른 가치 조합에 대한 연구를 통해 더 효율적인 알고리즘을 개발할 수 있을 것입니다. 또한, 다양한 가치 평가 시나리오에서의 복잡성을 비교하여 어떤 경우에 가장 효율적인 해결책이 제공될 수 있는지에 대한 통찰을 얻을 수 있을 것입니다.