A Strongly Polynomial-Time 7/9-Approximation Algorithm for Maximin Share Allocations

Abstract

We study maximin share (MMS) allocations of indivisible goods under nonnegative additive valuations. We give a strongly polynomial-time algorithm that allocates to every agent a bundle worth at least $7/9$ of her MMS, improving the previous $10/13$ guarantee of Heidari et al. (SODA 2026). Our analysis uses a dynamic witness-allocation framework that tracks the remaining goods from each agent's perspective throughout the allocation process. We obtain the computational result by iteratively refining upper bounds on MMS values, using only polynomially many runs of the allocation algorithm. Finally, we formalize and verify our main results in Lean 4.

arXiv commentWe have improved our result to a polynomial-time algorithm and reorganized the proof to be more structured