We present and analyze a simple, two-step algorithm to approximate the optimal solution of the sparse PCA problem. In the proposed approach, we first solve an ℓ 1 -penalized version of the NP-hard sparse PCA optimization problem and then we use a randomized rounding strategy to sparsify the resulting dense solution. Our main theoretical result guarantees an additive error approximation and provides a tradeoff between sparsity and accuracy. Extensive experimental evaluation indicates that the proposed approach is competitive in practice, even compared to state-of-the-art toolboxes such as Spasm.