Cardinality constrained submodular maximization for random streams

We consider the problem of maximizing submodular functions in single-pass streaming and secretaries-with-shortlists models, both with random arrival order. For cardinality constrained monotone functions, Agrawal, Shadravan, and Stein gave a single-pass (1-1/e-varepsilon)-approximation algorithm using only linear memory, but their exponential dependence on varepsilon makes it impractical even for varepsilon=0.1. We simplify both the algorithm and the analysis, obtaining an exponential improvement in the varepsilon-dependence (in particular, O(k/varepsilon) memory). Extending these techniques, we also give a simple (1/e-varepsilon)-approximation for non-monotone functions in O(k/varepsilon) memory. For the monotone case, we also give a corresponding unconditional hardness barrier of 1-1/e+varepsilon for single-pass algorithms in randomly ordered streams, even assuming unlimited computation. Finally, we show that the algorithms are simple to implement and work well on real world datasets.