EF1 and EFX Orientations

TL;DR

This paper investigates fair allocation of indivisible goods with orientations, providing a polynomial-time algorithm for EF1 and demonstrating NP-completeness for EFX in certain graph scenarios, along with a fixed-parameter tractable solution.

Contribution

It introduces a pseudopolynomial algorithm for EF1 orientations with monotone valuations and establishes NP-completeness for EFX orientations in specific graph cases, offering a fixed-parameter tractable algorithm.

Findings

EF1 orientations always exist with monotone valuations.

Algorithm finds EF1 orientations in various scenarios.

EFX orientation problem remains NP-complete in certain graph instances.

Abstract

We study the problem of finding fair allocations -- EF1 and EFX -- of indivisible goods with orientations. In an orientation, every agent gets items from their own predetermined set. For EF1, we show that EF1 orientations always exist when agents have monotone valuations, via a pseudopolynomial-time algorithm. This surprisingly positive result is the main contribution of our paper. We complement this result with a comprehensive set of scenarios where our algorithm, or a slight modification of it, finds an EF1 orientation in polynomial time. For EFX, we focus on the recently proposed graph instances, where every agent corresponds to a vertex on a graph and their allowed set of items consists of the edges incident to their vertex. It was shown that finding an EFX orientation is NP-complete in general. We prove that it remains intractable even when the graph has a vertex cover of size 8, or when we have a multigraph with only 10 vertices. We essentially match these strong negative results with a fixed-parameter tractable algorithm that is virtually the best someone could hope for.