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

TL;DR

This paper proves the existence of approximate envy-free allocations up to any k goods for additive agents, introduces polynomial-time algorithms for certain cases, and explores the complexity of EFkX graph orientations.

Contribution

It generalizes the 3PA algorithm to find approximate envy-free allocations for any k>2 and improves bounds for EF2X allocations among a fixed number of agents.

Findings

Existence of (k+1)/(k+2)-EFkX allocations for any number of agents and k>2.

Existence of 3/4-EF2X allocations for any number of agents.

NP-completeness of deciding EFkX graph orientations.

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 $k=1$.