We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents -- including graphs with vertex connectivity $1$ or $2$ and complete bipartite graphs -- as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.

翻译：我们研究不可分割物品的分配问题，这些物品构成无向图，并考察当要求每个智能体必须获得连通子图时的最坏情况福利损失。我们的研究重点同时涵盖平等主义福利和功利主义福利。具体而言，我们引入连通性的平等主义（相应地为功利主义）代价这一概念，该概念刻画了所有分配中最优平等主义（相应地为功利主义）福利与连通分配中最优福利之间的最坏情况比率。针对两类智能体的情形——包括顶点连通度为$1$或$2$的图以及完全二分图——我们给出了若干大类图的连通性代价的紧界或渐近紧界；同时针对路径图、星形图和环状图，我们给出了智能体数量可为任意值的一般情形下的相应界。