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=ε²) 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 α=(1 + α) − ε, where α is the approximation of the off-line algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to 1=2 − ε approximation (which is nearly optimal). If we postprocess with the state-of-the-art offline approximation algorithm, whose guarantee is α = 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(ε⁻²(logk + log(1 + α))) per element.