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arXiv · 2605.10371

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

Aris Filos-Ratsikas·Georgios Kalantzis·Fangxiao Wang
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Abstract

We study the problem of finding approximate envy-free allocations up to any $k$ goods ($\alpha$-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 $k=1$.

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Aris Filos-Ratsikas, Georgios Kalantzis, Fangxiao Wang. 2026-05-11. Approximate Envy-Free Allocations up to any $k$ Goods. https://arxiv.org/abs/2605.10371

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