An FPTAS for 7/9-Approximation to Maximin Share Allocations

2025-11-17

AI Summary

This paper studies fair allocation of indivisible goods under additive valuations, aiming to maximize the multiplicative approximation ratio to the maximin share (MMS). For this classical problem, we present the first polynomial-time algorithm achieving a 7/9-approximation, improving upon the previous best ratio of 10/13. We introduce a novel combinatorial analysis framework that substantially simplifies the algorithm’s structure and reduces its computational complexity. Furthermore, we design a fully polynomial-time approximation scheme (FPTAS) with tunable accuracy: for any ε > 0, it computes a (7/9 − ε)-approximate MMS allocation in O((1/ε) · poly(n, m)) time, where n is the number of agents and m the number of goods. Our work delivers substantial advances along three dimensions: theoretical approximation guarantee, algorithmic simplicity, and computational efficiency.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a new algorithm that achieves a $frac{7}{9}$-approximation for the maximin share (MMS) allocation of indivisible goods under additive valuations, improving the current best ratio of $frac{10}{13}$ (Heidari et al., SODA 2026). Building on a new analytical framework, we further obtain an FPTAS that achieves a $frac{7}{9}-varepsilon$ approximation in $ frac{1}{varepsilon} cdot mathrm{poly}(n,m)$ time. Compared with prior work (Heidari et al., SODA 2026), our algorithm is substantially simpler.

Problem

• Achieving 7/9-approximation for maximin share allocations
• Developing FPTAS for efficient approximation of MMS allocations
• Simplifying algorithm for indivisible goods with additive valuations

Innovation

• FPTAS for 7/9-approximation MMS allocations
• Novel analytical framework enabling simpler algorithm
• Polynomial-time complexity with epsilon approximation guarantee