## Abstract

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. The interest in MSP is twofold: on the one hand, there are many interesting applications of MSP, and on the other hand, there is strong hope that MSP admits O(1)-competitive algorithms, which is the claim of the well-known matroid secretary conjecture. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the sub modular matroid secretary problem (SMSP). The fact that sub modularity captures the property of diminishing returns, a very natural property for valuation functions, is a key reason for the interest in SMSP. So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most questions related to SMSP remained open, including whether SMSP may be substantially more difficult than MSP, and more generally, to what extend MSP and SMSP are related. Our goal is to address these points by presenting general black-box reductions from SMSP to MSP. In particular, we show that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O(1)-competitive algorithm for SMSP over the same matroid class. This implies that the matroid secretary conjecture is equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a particular matroid class, it suffices to consider MSP over the same matroid class. Using our reductions we obtain many first and improved O(1)-competitive algorithms for SMSP over various matroid classes by leveraging known algorithms for MSP. Moreover, our reductions imply an O(log log(rank))-competitive algorithm for SMSP, thus, matching the currently best asymptotic algorithm for MSP, and substantially improving on the previously best O(log(rank))-competitive algorithm for SMSP.

Original language | English |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |

Publisher | IEEE Computer Society |

Pages | 486-505 |

Number of pages | 20 |

ISBN (Electronic) | 9781467381918 |

DOIs | |

State | Published - 11 Dec 2015 |

Externally published | Yes |

Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: 17 Oct 2015 → 20 Oct 2015 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2015-December |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country/Territory | United States |

City | Berkeley |

Period | 17/10/15 → 20/10/15 |

### Bibliographical note

Publisher Copyright:© 2015 IEEE.

## Keywords

- matroids
- online algorithms
- secretary problem
- submodular functions