Exact MMS Allocations under Personalized Bivalued Valuations: Goods and Chores

2026-08-16

AI Summary

This study addresses the open problem regarding the existence of exact Maximin Share (MMS) allocations for items and chores under personalized bi-valued settings, providing the first affirmative resolution. By introducing a suite of combinatorial techniques—including quota restructuring, envelope relaxation, sparse extreme point construction, and flow rounding—this work theoretically proves that exact MMS allocations always exist in this context. Furthermore, it presents a polynomial-time algorithm to compute such allocations efficiently. These contributions overcome existing theoretical bottlenecks and establish a novel algorithmic framework with robust theoretical foundations for the fair allocation of mixed resources. Consequently, this research significantly advances the state of the art in discrete fair division by resolving a fundamental existence question and enabling practical computation.

Abstract

The maximin share (MMS) is a central fairness benchmark for allocating indivisible goods and chores. We study additive valuations in the personalized bivalued setting, where each agent assigns one of two agent-specific values to every item. Whether exact MMS allocations always exist in this setting has remained a major open question, as highlighted by Ebadian, Peters, and Shah and by Garg, Huang, and Segal-Halevi. We answer this question affirmatively: we prove that exact MMS allocations always exist for both goods and chores and can be computed in polynomial time. Our proof combines a quota-based reformulation with an envelope relaxation, a sparse extreme-point construction, and flow-based rounding that controls the total rounding loss.

Problem

• Maximin Share
• Personalized Bivalued Valuations
• Indivisible Goods and Chores
• Exact MMS Allocations
• Fair Division

Innovation

• Maximin Share
• Personalized Bivalued Valuations
• Envelope Relaxation
• Flow-based Rounding
• Sparse Extreme-point Construction