EF(X) Orientations: A Parameterized Complexity Perspective

TL;DR

This paper explores the computational complexity of envy-free orientations in graphs, introducing new tractability results, hardness proofs, and parameterized algorithms, especially for binary valuations and charity considerations.

Contribution

It initiates the study of EF orientations from a parameterized complexity perspective, providing new results and answering open questions about EFX orientations.

Findings

EF orientations are tractable in binary valuation cases.

Many results transfer to EFX orientations, improving previous work.

Algorithms for minimal edge removal to achieve EF orientations.

Abstract

The concept of fair orientations in graphs was introduced by Christodoulou, Fiat, Koutsoupias, and Sgouritsa in 2023, naturally modeling fair division scenarios in which resources are only contested by neighbors. In this model, vertices represent agents and undirected edges represent goods; edges have to be oriented towards one of their endpoints, i.e., allocated to one of their adjacent agents. Although EFX orientations (envy-free up to any good) have been extensively studied in this setting, EF orientations (envy-free) remain unexplored. In this work, we initiate their study, mostly under the lens of parameterized complexity, presenting various tractable cases, hardness results, and parameterizations. Our results concern both simple graphs and multigraphs. Interestingly, many of our results transfer to EFX orientations, thus complementing and improving upon previous work; notably, we answer an open question regarding the structural parameterized complexity of the latter problem on graphs of polynomially-bounded valuations. We also show that EF orientations are tractable in cases in which EFX orientations are not, particularly for binary valuations. Lastly, we consider charity in the orientation setting, establishing algorithms for finding the minimum amount of edges that have to be removed from a graph in order for EF(X) orientations to exist.