## Abstract

We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention, and the main focus is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraints) to allow 41C.5-Approximations online. It is known that the right-hand sides have to be i4.2 logm5 times the left-hand sides, where m is the number of constraints. In this paper, we give a primal-dual algorithm that achieves this bound for mixed packing/covering LPs. Our algorithms construct dual solutions using a regret-minimizing online learning algorithm in a black-box fashion, and use them to construct primal solutions. The adversarial guarantee that holds for the constructed duals helps us to take care of most of the correlations that arise in the algorithm; the remaining correlations are handled via martingale concentration and maximal inequalities. These ideas lead to conceptually simple and modular algorithms, which we hope will be useful in other contexts.

Original language | English (US) |
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Pages (from-to) | 1404-1431 |

Number of pages | 28 |

Journal | Mathematics of Operations Research |

Volume | 41 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2016 |

## Keywords

- Experts Algorithm
- Linear Programming
- Online Algorithms
- Secretary Problem

## ASJC Scopus subject areas

- General Mathematics
- Computer Science Applications
- Management Science and Operations Research