## Abstract

We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi) streaming algorithm that uses roughly O(k=ε^{2}) memory, where k is the size constraint. At the end of the stream, our algorithm postprocesses its data structure using any off-line algorithm for submodular maximization and obtains a solution whose approximation guarantee is α=(1 + α) − ε, where α is the approximation of the off-line algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to 1=2 − ε approximation (which is nearly optimal). If we postprocess with the state-of-the-art offline approximation algorithm, whose guarantee is α = 0:385, we obtain a 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715. It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for nonmonotone submodular maximization, and enjoys a fast update time of O(ε^{−2}(logk + log(1 + α))) per element.

Original language | English |
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Pages (from-to) | 2667-2690 |

Number of pages | 24 |

Journal | Mathematics of Operations Research |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2022 |

Externally published | Yes |

### Bibliographical note

Funding Information:Funding: The work of N. Alaluf and M. Feldman was supported in part by the Israel Science Foundation [Grant 1357/16]. The work of A. Ene and A. Suh was supported in part by the National Science Fun-dation (NSF) [CAREER Grant CCF-1750333 and Grants CCF-1718342 and III-1908510]. The work of H. L. Nguyen was supported in part by the NSF [CAREER Grant CCF-1750716 and Grant CCF-1909314].

Publisher Copyright:

Copyright: © 2022 INFORMS.

## Keywords

- cardinality constraint
- semi-streaming algorithms
- submodular maximization