EFX Allocations Exist on Triangle-Free Multi-Graphs

2025-12-25

AI Summary

This paper investigates the existence of EFX (envy-free up to any good) allocations under the multi-graph valuation model, focusing specifically on triangle-free multi-graphs—i.e., multi-graphs containing no 3-cycles. In this model, multiple agents may assign nonzero valuations to the same item, enabling richer, more realistic representations of collaborative or shared valuations. The authors establish, for the first time, that an EFX allocation always exists on any triangle-free multi-graph. They further devise the first pseudo-polynomial-time constructive algorithm for computing such an allocation. Under the additional assumption of cancelable valuations, the algorithm is refined to run in truly polynomial time. This work significantly broadens the scope of known EFX existence results, and stands among the rare general-purpose guarantees that hold for an arbitrary number of agents, are explicitly constructive, and admit polynomial-time computation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central open question in this area is the existence of EFX allocations-allocations in which any envy of any agent i towards any agent j vanishes upon the removal of any single good from j's bundle. Establishing the existence of such allocations has proven notoriously difficult in general, but progress has been made for restricted valuation classes. Christodoulou et al. [2023] proved existence for graphical valuations, where goods correspond to edges in a graph, agents to nodes, and each agent values only incident edges. The graph was required to be simple, i.e., for any pair of agents, there could be at most one good that both agents value. The problem remained open, however, for multi-graph valuations, where for a pair of agents several goods may have value to both. In this setting, Sgouritsa and Sotiriou [2025] established existence whenever the shortest cycle with non-parallel edges has length at least six, while Afshinmehr et al. [2025] proved existence when the graph contains no odd cycles. In this paper, we strengthen these results by proving that EFX allocations always exist in multi-graphs that contain no cycle of length three. Assuming monotone valuations, we further provide a pseudo-polynomial time algorithm for computing such an allocation, which runs in polynomial time when agents have cancelable valuations, a strict superclass of additive valuation functions. Accordingly, our results stand as one of the only cases where EFX allocations exist for an arbitrary number of agents.

Problem

• Proving EFX allocations exist in triangle-free multi-graphs
• Addressing envy-free allocations with indivisible goods among agents
• Providing a pseudo-polynomial algorithm for computing such allocations

Innovation

• Proves EFX existence on triangle-free multi-graphs
• Provides pseudo-polynomial algorithm for monotone valuations
• Runs in polynomial time for cancelable valuations