New algorithm guarantees envy-free allocations for any number of agents

A team of computer scientists—Filos-Ratsikas, Kalantzis, and Wang—has published a breakthrough in fair division theory, tackling the problem of allocating indivisible goods to agents with additive valuations while minimizing envy. Their paper, now on arXiv, introduces a generalized version of the 3PA algorithm (originally by Amanatidis et al., 2024) that guarantees approximate envy-free allocations up to any k goods, denoted α-EFkX. For any k > 2, they achieve a ratio of (k+1)/(k+2), meaning each agent gets at least that fraction of their value for the best bundle they could envy after removing k items from others.

This result yields immediate corollaries: 3/4-EF2X allocations exist for any number of agents—a notable improvement over the previous state-of-the-art, which only guaranteed 2/3-EFX for up to 7 agents. The authors extend this further by devising an algorithm that achieves 2/3-EFX for 8 agents, closing the gap slightly. On the negative side, they prove that EFkX graph orientations (a related model) do not always exist, and deciding their existence is NP-complete, generalizing prior work for k=1. These findings have practical implications for algorithmic fairness in resource allocation, from cloud computing to multi-agent AI systems.