We consider a cost sharing problem on a weighted undirected graph, where all the nodes want to connect to a special node called source, and they need to share the total cost (weights) of the used edges. Each node except for the source has a private valuation of the connection, and it may block others' connections by strategically cutting its adjacent edges to reduce its cost share, which may increase the total cost. We aim to design mechanisms to prevent the nodes from misreporting their valuations and cutting their adjacent edges. We first show that it is impossible for such a mechanism to further satisfy budget balance (cover the total cost) and efficiency (maximize social welfare). Then, we design two feasible cost sharing mechanisms that incentivize each node to offer all its adjacent edges and truthfully report its valuation, and also satisfy either budget balance or efficiency.

翻译：我们考虑一个加权无向图上的成本分摊问题，其中所有节点都希望连接到称为源节点的特殊节点，并且它们需要分摊所使用边的总成本（权重）。除源节点外，每个节点对连接拥有私有估值，并可能通过策略性地切断其相邻边来阻止其他节点的连接，从而减少自身分摊的成本，但这可能导致总成本增加。我们旨在设计机制以防止节点虚报其估值或切断其相邻边。我们首先证明，此类机制不可能同时满足预算平衡（覆盖总成本）和效率（最大化社会总福利）。随后，我们设计了两种可行的成本分摊机制，既能激励每个节点提供其所有相邻边并如实报告其估值，又能分别满足预算平衡或效率中的一项。