### Abstract

Suppose a set of requests arrives online: each request gives some value v_{i} if accepted, but requires using some amount of each of d resources. Our cost is a convex function of the vector of total utilization of these d resources. Which requests should be accept to maximize our profit, i.e., the sum of values of the accepted demands, minus the convex cost? We consider this problem in the random-order a.k.a. secretary model, and show an O(d)competitive algorithm for the case where the convex cost function is also supermodular. If the set of accepted demands must also be independent in a given matroid, we give an O(d^{3}α)-competitive algorithm for the supermodular case, and an improved O(d^{2}α) if the convex cost function is also separable. Here α is the competitive ratio of the best algorithm for the submodular secretary problem. These extend and improve previous results known for this problem. Our techniques are simple but use powerful ideas from convex duality, which give clean interpretations of existing work, and allow us to give the extensions and improvements.

Original language | English (US) |
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Title of host publication | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |

Editors | Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770767 |

DOIs | |

State | Published - Jul 1 2018 |

Event | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic Duration: Jul 9 2018 → Jul 13 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 107 |

ISSN (Print) | 1868-8969 |

### Other

Other | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |
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Country | Czech Republic |

City | Prague |

Period | 7/9/18 → 7/13/18 |

### Keywords

- Convex duality
- Online algorithms
- Random order
- Secretary problem

### ASJC Scopus subject areas

- Software

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## Cite this

*45th International Colloquium on Automata, Languages, and Programming, ICALP 2018*[71] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 107). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ICALP.2018.71