Link-bandwidth utilization and flow information are obviously critical for numerous network management tasks. The problem of efficiently monitoring the network flowing based on flow-conservation could be reduced to Weak Vertex Cover problem, which is NP-hard. In this paper, using the primal-dual method, we give an approximation algorithm with approximation ratio 2 to solve the problem. It is a near-optimal algorithm as it is very difficult to get an approximation algorithm with approximation ratio lower than 2 for Weak Vertex Cover problem. The effectiveness of our monitoring algorithm is validated by simulations evaluation over a wide range of network topologies. The practices indicate that our work is valuable to solve Weak Vertex Cover problem and its application in network management.