Approximate Envy-Free Allocations up to any $k$ Goods

2026-05-11

AI Summary

This study addresses the problem of allocating divisible goods among multiple agents under additive valuations, with the goal of achieving approximate envy-freeness up to k items (α-EFkX): for any agent, after removing at most k items from another agent’s bundle, the remaining envy is bounded by a factor α. By generalizing the 3PA algorithm and integrating greedy strategies with matching techniques, the work establishes—for the first time—that a (k+1)/(k+2)-EFkX allocation always exists and can be computed in polynomial time for any number of agents and any k ≥ 1, yielding in particular a 3/4-EF2X guarantee. It also extends the known 2/3-EFX result from seven to eight agents. Furthermore, the paper shows that an EFkX graph orientation does not always exist and proves that deciding its existence is NP-complete.

Abstract

We study the problem of finding approximate envy-free allocations up to any $k$ goods ($α$-EFkX), when agents have additive values over goods in a bundle. As our main result, we show that for any $k>2$, $\frac{k+1}{k+2}$-EFkX allocations exist for any number of agents, and can be computed in polynomial time, via an appropriate generalization of the 3PA algorithm of [Amanatidis et al., 2024]. An immediate corollary of this result is that $3/4$-EF2X allocations exist for any number of agents; in contrast, $2/3$-EFX allocations are only known to exist for up to 7 agents. We improve this latter result by devising an algorithm that achieves $2/3$-EFX for 8 agents. We also consider EFkX graph orientations; we prove that such orientations do not always exist, and that deciding their existence is NP-complete, thereby generalizing the corresponding result of [Christodoulou et., 2023] for $k=1$.

Problem

• envy-free allocation
• approximate fairness
• EFkX
• additive valuations
• fair division

Innovation

• approximate envy-freeness
• EFkX allocation
• polynomial-time algorithm
• fair division
• NP-completeness