Making a Sieve Random: Improved Semi-Streaming Algorithm for Submodular Maximization under a Cardinality Constraint
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In this paper we consider the problem of maximizing a non-negative submodular function subject to a cardinality constraint in the data stream model. Previously, the best known algorithm for this problem was a 5.828-approximation semi-streaming algorithm based on a local search technique (Feldman et al., 2018). For the special case of this problem in which the objective function is also monotone, the state-of-the-art semi-streaming algorithm is an algorithm known as Sieve-Streaming, which is based on a different technique (Badanidiyuru, 2014). Adapting the technique of Sieve-Streaming to non-monotone objective functions has turned out to be a challenging task, which has so far prevented an improvement over the local search based 5.828-approximation. In this work, we overcome the above challenge, and manage to adapt Sieve-Streaming to non-monotone objective functions by introducing a "just right" amount of randomness into it. Consequently, we get a semi-streaming polynomial time 4.282-approximation algorithm for non-monotone objectives. Moreover, if one allows our algorithm to run in super-polynomial time, then its approximation ratio can be further improved to 3 + ε.
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