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.

翻译：我们考虑将一组异构可分割资源公平分配给具有不同偏好的智能体的问题。重点研究资源对应于连通图边集、每个智能体必须被分配该图的一个连通子图，且所考虑的公平性概念为经典的无嫉妒性。该问题是NP完全的，我们从两个自然复杂度度量——智能体数量与图边数——出发分析其复杂性。尽管即便在仅含2个智能体的实例中该问题仍保持NP难性，但我们基于图的结构性质，给出了当智能体数量为常数时问题复杂性的二分法刻画。针对后一种情况，当图具有常数条边时，我们设计了一个多项式时间算法。