Improved Approximate EFX Guarantees for Multigraphs

TL;DR

This paper improves the guarantees for fair allocation in multigraph models by proving the existence of a better FX allocation for additive (2,) bounded valuations, advancing fairness guarantees in complex valuation settings.

Contribution

It introduces a new FX guarantee for additive (2,) bounded valuations, surpassing previous bounds in multigraph fair division models.

Findings

Established a FX allocation with a / bound.

Improved previous FX guarantee from /3 to 1/.

Applicable to multigraph models with bounded valuations.

Abstract

In recent years, a new line of work in fair allocation has focused on EFX allocations for \((p, q)\)-bounded valuations, where each good is relevant to at most \(p\) agents, and any pair of agents share at most \(q\) relevant goods. For the case \(p = 2\) and \(q = \infty\), such instances can be equivalently represented as multigraphs whose vertices are the agents and whose edges represent goods, each edge incident to exactly the one or two agents for whom the good is relevant. A recent result of \citet{amanatidis2024pushing} shows that for additive $(2,\infty)$ bounded valuations, a \((\nicefrac{2}{3})\)-EFX allocation always exists. In this paper, we improve this bound by proving the existence of a \((\nicefrac{1}{\sqrt{2}})\)-\(\efx\) allocation for additive \((2,\infty)\)-bounded valuations.