We consider a cost sharing problem to connect all nodes in a weighted undirected graph, where the weight of each edge represents the cost to use the edge for the connectivity and the cost has to be shared among all connected nodes. There is one node called the source to which all the other nodes want to connect and it does not share the costs of the connectivity. As a node may need to go through other nodes to reach the source, the intermediate nodes may behave strategically to block the connection by cutting the edges adjacent to them. To prevent such strategical behavior, we design cost sharing mechanisms to incentivize all nodes not to cut any edge so that we can minimize the total cost for connecting all the nodes.

翻译：我们考虑一个加权无向图中的成本分摊问题，旨在连接所有节点。图中每条边的权重表示使用该边实现连通性的成本，该成本需由所有已连接节点共同分摊。存在一个称为源节点的特殊节点，所有其他节点均需与之建立连接，且源节点不参与连通性成本的分摊。由于节点可能需要经由其他节点到达源节点，中间节点可能采取策略性行为，通过切断与其相邻的边来阻碍连接。为防止此类策略性行为，我们设计了成本分摊机制，激励所有节点不切断任何边，从而最小化连接所有节点的总成本。