Many real-world datasets can be represented in the form of a graph whose edge
weights designate similarities between instances. A discrete Gaussian random
field (GRF) model is a finite-dimensional Gaussian process (GP) whose prior
covariance is the inverse of a graph Laplacian. Minimizing the trace of the
predictive covariance Sigma (V-optimality) on GRFs has proven successful in
batch active learning classification problems with budget constraints. However,
its worst-case bound has been missing. We show that the V-optimality on GRFs as
a function of the batch query set is submodular and hence its greedy selection
algorithm guarantees an (1-1/e) approximation ratio. Moreover, GRF models have
the absence-of-suppressor (AofS) condition. For active survey problems, we
propose a similar survey criterion which minimizes 1'(Sigma)1. In practice,
V-optimality criterion performs better than GPs with mutual information gain
criteria and allows nonuniform costs for different nodes