### Abstract

Let X = {1,2,..., n} be a ground set of n elements, and let S be a family of subsets of X, |S| = m, with a positive cost c _{S} associated with each S ∈ S. Consider the following online version of the set cover problem, described as a game between an algorithm and an adversary. An adversary gives elements to the algorithm from X one by one. Once a new element is given, the algorithm has to cover it by some set of S containing it. We assume that the elements of X and the members of S are known in advance to the algorithm; however, the set X' ⊆ X of elements given by the adversary is not known in advance to the algorithm. (In general, X' may be a strict subset of X.) The objective is to minimize the total cost of the sets chosen by the algorithm. Let C denote the family of sets in S that the algorithm chooses. At the end of the game the adversary also produces (offline) a family of sets C _{OPT} that covers X'. The performance of the algorithm is the ratio between the cost of C and the cost of C _{OPT}. The maximum ratio, taken over all input sequences, is the competitive ratio of the algorithm. We present an O(log m log n) competitive deterministic algorithm for the problem and establish a nearly matching Ω (log n log m/log log m+log log n) lower bound for all interesting values of m and n. The techniques used are motivated by similar techniques developed in computational learning theory for online prediction (e.g., the WINNOW algorithm) together with a novel way of converting a fractional solution into a deterministic online algorithm.

Original language | English (US) |
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Pages (from-to) | 361-370 |

Number of pages | 10 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2009 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Keywords

- Competitive factor
- Online
- Set cover

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## Cite this

*SIAM Journal on Computing*,

*39*(2), 361-370. https://doi.org/10.1137/060661946