## Abstract

A covering integer program (CIP) is a mathematical program of the form min{c^{⊤}x | Ax ≥ 1,0 ≤ x ≤ u, x ∈ ℤ^{n}}, where all entries in A, c , u are nonnegative. In the online setting, the constraints (i.e., the rows of the constraint matrix A) arrive over time, and the algorithm can only increase the coordinates of x to maintain feasibility. As an intermediate step, we consider solving the covering linear program (CLP) online, where the integrality constraints are dropped. Our main results are (a) an O(log k)-competitive online algorithm for solving the CLP, and (b) an O(log k·logℓ)-competitive randomized online algorithm for solving the CIP. Here k ≤ n and ℓ ≤ m respectively denote the maximum number of nonzero entries in any row and column of the constraint matrix A. Our algorithm is based on the online primal-dual paradigm, where a novel ingredient is to allow dual variables to increase and decrease throughout the course of the algorithm. It is known that this result is the best possible for polynomial-time online algorithms, even in the special case of set cover (where all entries in A, c , and u are 0 or 1.

Original language | English (US) |
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Pages (from-to) | 998-1011 |

Number of pages | 14 |

Journal | Mathematics of Operations Research |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1 2014 |

## Keywords

- Covering integer programs
- Linear programming
- Online algorithms

## ASJC Scopus subject areas

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