Monitoring end-to-end paths in an overlay network is essential for evaluating end-system performance and for troubleshooting anomalous behavior. However, conducting measurements between all pairs of overlay nodes can be cumbersome and expensive, especially in a large network. In this paper, we take a different, approach and explore ail additional degree of freedom, namely, monitoring native links. We allow native link measurements, as well as end-to-end overlay measurements. in order to minimize the total cost of monitoring the network. We formulate an optimization problem that, when solved, identifies the optimal set of native and overlay links to monitor, and a feasible sequence of arithmetic operations to perforin for inferring characteristics of the overlay links that are not monitored directly. We use simulations to investigate how various topological properties may affect; the best; monitoring strategy. We also conduct measurements over the PlanetLab network to quantify the accuracy of different monitoring strategies.