Feature selection is an important task in order to achieve better generalizability in high dimensional learning, and structure learning of Markov random fields (MRFs) can automatically discover the inherent structures underlying complex data. Both problems can be cast as solving an ℓ1-norm regularized parameter estimation problem. To solve such an ℓ1-regularized estimation problem, the existing Grafting [16] method can avoid doing inference on dense graphs in structure learning by incrementally selecting new features. However, Grafting performs a greedy step of optimizing over free parameters once new features are included. This greedy strategy results in low efficiency when parameter learning is itself non-trivial, such as in MRFs, in which parameter learning depends on an expensive subroutine to calculate gradients. The complexity of calculating gradients in MRFs is typically exponential to the size of maximal cliques. In this paper, we present a fast algorithm called Grafting- Light to solve the ℓ1-norm regularized maximum likelihood estimation of MRFs for efficient feature selection and structure learning. Grafting-Light iteratively performs one-step of orthant-wise gradient descent over free parameters and selects new features. This lazy strategy is guaranteed to converge to the global optimum and can effectively select significant features. On both synthetic and real data sets, we show that Grafting-Light is much more efficient than Grafting for both feature selection and structure learning, and performs comparably with the optimal batch method for feature selection but is much more efficient and accurate for structure learning of MRFs.