  In a cost sharing problem on a weighted undirected graph, all other nodes
want to connect to the source node for some service. Each edge has a cost
denoted by a weight and all the connected nodes should share the total cost for
the connectivity. The goal of the existing solutions (e.g. folk solution and
cycle-complete solution) is to design cost sharing rules with nice properties,
e.g. budget balance and cost monotonicity. However, they did not consider the
cases that each non-source node has a budget which is the maximum it can pay
for its cost share and may cut its adjacent edges to reduce its cost share. In
this paper, we design two cost sharing mechanisms taking into account the
nodes' budgets and incentivizing all nodes to report all their adjacent edges
so that we can minimize the total cost for the connectivity.
