### Abstract

Suppose we are given a submodular function f over a set of elements, and we want to maximize its value subject to certain constraints. Good approximation algorithms are known for such problems under both monotone and non-monotone submodular functions. We consider these problems in a stochastic setting, where elements are not all active and we only get value from active elements. Each element e is active independently with some known probability pe, but we don't know the element's status a priori: we find it out only when we probe the element e. Moreover, the sequence of elements we probe must satisfy a given prefix-closed constraint, e.g., matroid, orienteering, deadline, precedence, or any downward-closed constraint. In this paper we study the gap between adaptive and non-adaptive strategies for f being a submodular or a fractionally subadditive (XOS) function. If this gap is small, we can focus on finding good non-adaptive strategies instead, which are easier to find as well as to represent. We show that the adaptivity gap is a constant for monotone and non-monotone submodular functions, and logarithmic for XOS functions of small width. These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic optimization problems.

Original language | English (US) |
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Title of host publication | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |

Editors | Philip N. Klein |

Publisher | Association for Computing Machinery |

Pages | 1688-1702 |

Number of pages | 15 |

ISBN (Electronic) | 9781611974782 |

DOIs | |

State | Published - Jan 1 2017 |

Externally published | Yes |

Event | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain Duration: Jan 16 2017 → Jan 19 2017 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 0 |

### Other

Other | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |
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Country | Spain |

City | Barcelona |

Period | 1/16/17 → 1/19/17 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

*28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017*(pp. 1688-1702). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 0). Association for Computing Machinery. https://doi.org/10.1137/1.9781611974782.111