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In this work we give two new algorithms that use similar techniques for(non-monotone) submodular function maximization subject to a cardinalityconstraint.  The first is an offline fixed parameter tractable algorithm that guarantees a$0.539$-approximation for all non-negative submodular functions.  The second algorithm works in the random-order streaming model. It guaranteesa $(1/2+c)$-approximation for symmetric functions, and we complement it byshowing that no space-efficient algorithm can beat $1/2$ for asymmetricfunctions. To the best of our knowledge this is the first provable separationbetween symmetric and asymmetric submodular function maximization.

Maximizing Non-Monotone Submodular Functions over Small Subsets: Beyond $1/2$-Approximation