We study the allocation of indivisible goods 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 the connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for various large classes of graphs when there are two agents as well as for paths, stars and cycles in the general case. Many of our results are supplemented with algorithms which find connected allocations with a welfare guarantee corresponding to the price of connectivity.

翻译：本文研究由不可分割商品构成的连通图分配问题，并探讨当要求每个智能体必须获得连通子图时，最坏情况下的福利损失。我们同时关注平等福利与功利福利。具体而言，我们引入了连通性的平等（或功利）代价概念，该概念刻画了所有分配中的最优平等（或功利）福利与连通分配中的最优福利之间的最坏情况比率。针对两个智能体以及一般情况下的路径图、星图和环图，我们给出了多种广泛图类上连通性代价的紧界或渐近紧界。许多结果辅以相应的算法，这些算法能够找到具有与连通性代价对应的福利保障的连通分配。