Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints.

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two single-pass (semi-)streaming algorithms that use $\tilde{O}(k)\cdot\mathrm{poly}(1/\varepsilon)$ memory, where $k$ is the size constraint. At the end of the stream, both our algorithms post-process their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is $\frac{\alpha}{1+\alpha}-\varepsilon$, where $\alpha$ is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to $\frac{1}{2}-\varepsilon$ approximation (which is nearly optimal). If we post-process with the algorithm of Buchbinder and Feldman (Math of OR 2019), that achieves the state-of-the-art offline approximation guarantee of $\alpha=0.385$, we obtain $0.2779$-approximation in polynomial time, improving over the previously best polynomial-time approximation of $0.1715$ due to Feldman et al. (NeurIPS 2018). One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest.