# The Complexity of Envy-Free Graph Cutting

# The Complexity of Envy-Free Graph Cutting

## Argyrios Deligkas, Eduard Eiben, Robert Ganian, Thekla Hamm, Sebastian Ordyniak

Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence

Main Track. Pages 237-243.
https://doi.org/10.24963/ijcai.2022/34

We consider the problem of fairly dividing a set of heterogeneous divisible resources among agents with different preferences. We focus on the setting where the resources correspond to the edges of a connected graph, every agent must be assigned a connected piece of this graph, and the fairness notion considered is the classical envy freeness. The problem is NP-complete, and we analyze its complexity with respect to two natural complexity measures: the number of agents and the number of edges in the graph. While the problem remains NP-hard even for instances with 2 agents, we provide a dichotomy characterizing the complexity of the problem when the number of agents is constant based on structural properties of the graph. For the latter case, we design a polynomial-time algorithm when the graph has a constant number of edges.

Keywords:

Agent-based and Multi-agent Systems: Resource Allocation

Agent-based and Multi-agent Systems: Computational Social Choice