Continuously monitoring the link performance is important to network diagnosis. Recently, active probes sent between end systems are widely used to monitor the link performance. In this paper, we address the problem of minimizing the probing cost and achieving identifiability in link monitoring. Given a set of links to monitor, our objective is to select as few probing paths as possible to cover all of them, and the selected probing paths can uniquely identify all identifiable links being monitored. We propose an algorithm based on the linear system model to find out all sets of probing paths that can uniquely identify an identifiable link. We extend the bipartite model to reflect the relation between a set of probing paths and the link that can be uniquely identified. Through the extended bipartite model, our optimization problem is transformed into the classic set cover problem, which is NP-hard. Therefore, we propose a heuristic based algorithm to greedily select the probing paths. Our method eliminates two types of redundant probing paths, i.e., those that can be replaced by others and those that cannot be used to achieving identifiability. Simulations based on real network topologies show that our approach can achieve identifiability with very low probing cost. Compared with prior work, our method is more general and has better performance.