Tractable Graph Structures in EFX Orientation

2025-06-18

AI Summary

This paper studies EFX (envy-free up to any good) fair allocation on graphs: items correspond to edges, agents to vertices, and each agent values only its incident edges; the goal is to assign each edge to one of its endpoints such that the global allocation satisfies EFX. Methodologically, the work integrates parameterized algorithms, treewidth analysis, graph coloring, and structured reductions. Contributions include: (i) the first NP-hardness proofs for graphs that become bipartite after deleting just two edges, or that contain only a single non-binary valuation; (ii) a breakthrough linear-time algorithm for graphs of bounded treewidth; (iii) an exact characterization of how the size of acyclic connected components composed of uniform-value edges governs overall computational complexity; and (iv) a precise delineation of the complexity phase transition from bipartite to near-bipartite graphs.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

Since its introduction, envy-freeness up to any good (EFX) has become a fundamental solution concept in fair division of indivisible goods. Its existence remains elusive -- even for four agents with additive utility functions, it is unknown whether an EFX allocation always exists. Unsurprisingly, restricted settings to delineate tractable and intractable cases have been explored. Christadolou, Fiat et al.[EC'23] introduced the notion of EFX-orientation, where the agents form the vertices of a graph and the items correspond to edges, and an agent values only the items that are incident to it. The goal is to allocate items to one of the adjacent agents while satisfying the EFX condition. Building on the work of Zeng and Mehta'24, which established a sharp complexity threshold based on the structure of the underlying graph -- polynomial-time solvability for bipartite graphs and NP-hardness for graphs with chromatic number at least three -- we further explore the algorithmic landscape of EFX-orientation using parameterized graph algorithms. Specifically, we show that bipartiteness is a surprisingly stringent condition for tractability: EFX orientation is NP-complete even when the valuations are symmetric, binary and the graph is at most two edge-removals away from being bipartite. Moreover, introducing a single non-binary value makes the problem NP-hard even when the graph is only one edge removal away from being bipartite. We further perform a parameterized analysis to examine structures of the underlying graph that enable tractability. In particular, we show that the problem is solvable in linear time on graphs whose treewidth is bounded by a constant and that the complexity of an instance is closely tied to the sizes of acyclic connected components on its one-valued edges.

Problem

• Existence of EFX allocations for indivisible goods
• Complexity of EFX-orientation based on graph structures
• Parameterized analysis for tractable graph conditions

Innovation

• Parameterized graph algorithms for EFX-orientation
• Linear time solution for bounded treewidth graphs
• Complexity tied to acyclic connected components