Approximate Envy-Free Allocations up to any k Goods
For any k>2, (k+1)/(k+2)-EFkX allocations always exist and are computable in polynomial time, yielding 3/4-EF2X for any number of agents and 2/3-EF X for eight agents.
Published 2026Paris Poster Session 3 · Thu, Dec 10, 12:30 PM–2:30 PM local time · Paris Poster HallarXiv ↗OpenReview ↗
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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 $k=1$.