An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality Constraint

We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi) streaming algorithm that uses roughly $O(k/{\epsilon }^2)$ memory, where k is the size constraint. At the end of the stream, our algorithm postprocesses its data structure using any off-line algorithm for submodular maximization and obtains a solution whose approximation guarantee is $\alpha /(1+\alpha )-\epsilon$, where α is the approximation of the off-line algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to $1/2-\epsilon$ approximation (which is nearly optimal). If we postprocess with the state-of-the-art offline approximation algorithm, whose guarantee is $\alpha =0.385$, we obtain a 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715. It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for nonmonotone submodular maximization, and enjoys a fast update time of $O({\epsilon }^{\mathrm{-2}}(\log k+\log (1+\alpha )))$ per element.