# Simultaenous Sieves: A Deterministic Streaming Algorithm for Non-Monotone Submodular Maximization

In this work, we present a combinatorial, deterministic single-pass streaming algorithm for the problem of maximizing a submodular function, not necessarily monotone, with respect to a cardinality constraint (SMCC). In the case the function is monotone, our algorithm reduces to the optimal streaming algorithm of Badanidiyuru et al. (2014). In general, our algorithm achieves ratio α / (1 + α) - ε, for any ε > 0, where α is the ratio of an offline (deterministic) algorithm for SMCC used for post-processing. Thus, if exponential computation time is allowed, our algorithm deterministically achieves nearly the optimal 1/2 ratio. These results nearly match those of a recently proposed, randomized streaming algorithm that achieves the same ratios in expectation. For a deterministic, single-pass streaming algorithm, our algorithm achieves in polynomial time an improvement of the best approximation factor from 1/9 of previous literature to ≈ 0.2689.

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