Polynomial-Time Algorithms for Fair Orientations of Chores

TL;DR

This work resolves the complexity of fair orientations for chores on graphs and multigraphs, proving polynomial-time algorithms for EF1 and $EFX_0$ orientations when only chores are present, even with self-loops, and establishing NP-completeness for the multigraph case. The authors introduce a layered reduction framework where EFX$_0$-Orientation reduces to a combination of $2$-SAT and PD-Vertex-Cover problems, via a subdivision-based transformation that handles non-objective edges, and a PD-Vertex-Cover reduction that integrates auxiliary constraints. A key insight is a clear separation between chore and goods settings: EF1 is always tractable for chores on graphs, in contrast to goods, and $EFX_0$ is tractable for chores while its goods counterpart remains NP-complete; the results extend to additive utilities. The paper thus provides efficient algorithms for large graphs, clarifies the landscape of fair-division with chores, and identifies natural hardness borders in multigraphs, enriching the theory of fair orientations and their algorithmic applications.

Abstract

This paper addresses the problem of finding fair orientations of graphs of chores, in which each vertex corresponds to an agent, each edge corresponds to a chore, and a chore has zero marginal utility to an agent if its corresponding edge is not incident to the vertex corresponding to the agent. Recently, Zhou et al. (IJCAI, 2024) analyzed the complexity of deciding whether graphs containing a mixture of goods and chores have EFX orientations, and conjectured that deciding whether graphs containing only chores have EFX orientations is NP-complete. We resolve this conjecture by giving polynomial-time algorithms that find EF1 and EFX orientations of graphs containing only chores if they exist, even if there are self-loops. Remarkably, our result demonstrates a surprising separation between the case of goods and the case of chores, because deciding whether graphs containing only goods have EFX orientations was shown to be NP-complete by Christodoulou et al. (EC, 2023). In addition, we show the EF1 and EFX orientation problems for multigraphs to be NP-complete.

Figures (5)

• Figure 1: The graph of an objective instance. Solid (resp. dashed) edges denote negative (resp. dummy) edges. The set of black (resp. white) vertices induces one negative component.
• Figure 2: The reductions our algorithms represent. Each node on the left is a decision problem. A directed edge between nodes is a reduction and is labelled with the relevant algorithm.
• Figure 3: Subdivision of $e_{ij}$ during the construction of $G^o$ by FindEFXOrientation. The labels above an edge indicate the utility the edge has to its two endpoints. We write $\beta \coloneqq u_j(e_{ij})$ for clarity.
• Figure 4: The three possibilities involving the fake edges $e_{ik}, e_{jk}$ in the EFX$_0$ orientation $\pi^o$ of $G^o$.
• Figure 5: The multigraph $G$ in Theorem \ref{['thm:main-no-self-loop']}. Here, $T \coloneqq \sum_{i \in [k]} s_i$.

Theorems & Definitions (28)

• Definition 1: EF1
• Definition 2: EFX$_0$
• Definition 3: PD-Vertex-Cover
• Definition 4: 2SAT
• Definition 5: Partition
• Proposition 1
• proof
• Proposition 2
• Theorem 3
• proof