We investigate the problem of placing a given number of monitors in a communication network to identify the maximum number of link metrics from end-to-end measurements between monitors, assuming that link metrics are additive, and measurement paths cannot contain cycles. Motivated by our previous result that complete identification of all link metrics can require a large number of monitors, we focus on partial identification using a limited number of monitors. The basis to our solution is an efficient algorithm for determining all identifiable links for a given monitor placement. Based on this algorithm, we develop a polynomial-time greedy algorithm to incrementally place monitors such that each newly placed monitor maximizes the number of additional identifiable links. We prove that the proposed algorithm is optimal for 2-vertex-connected networks, and demonstrate that it is near-optimal for several real ISP topologies that are not 2-vertex-connected. Our solution provides a quantifiable tradeoff between level of identifiability and available monitor resources.