The growing amount of applications that generate vast amount of data in short time scales render the problem of partial monitoring, coupled with prediction, a rather fundamental one. We study the aforementioned canonical problem under the context of large-scale monitoring of communication networks. We consider the problem of selecting the "best" subset of links so as to optimally predict the quantity of interest at the remaining ones. This is a well know NP-hard problem, and algorithms seeking the exact solution are prohibitively expensive. We present a number of approximation algorithms that: 1) their computational complexity gains a significant improvement over existing greedy algorithms; 2) exploit the geometry of principal component analysis, which also helps us establish theoretical bounds on the prediction error; 3) are amenable for randomized implementation and execution in parallel or distributed fashion, a process that often yields the exact solution. The new algorithms are demonstrated and evaluated using real-world network data.