### Abstract

In this paper, we consider the weighted online set k-multicover problem. In this problem, we have a universe V of elements, a family S of subsets of V with a positive real cost for every S ∈ S, and a "coverage factor" (positive integer) k. A subset {i_{0}, i_{1}, ...} ⊆ V of elements are presented online in an arbitrary order. When each element i_{p} is presented, we are also told the collection of all (at least k) sets S_{ip} ⊆ S and their costs to which i_{p} belongs and we need to select additional sets from S_{ip} if necessary such that our collection of selected sets contains at leastk sets that contain the element i_{p}. The goal is to minimize the total cost of the selected sets.^{1}1Our algorithm and competitive ratio bounds can be extended to the case when a set can be selected at most a prespecified number of times instead of just once; we do not report these extensions for simplicity and also because they have no relevance to the biological applications that motivated our work. In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of [N. Littlestone, Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm, Machine Learning 2 (1988) 285-318]. This algorithm generalizes and improves some earlier results in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, A general approach to online network optimization problems, in: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, 2004, pp. 570-579; N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100-105]. We also discuss lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100-105].

Original language | English (US) |
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Pages (from-to) | 54-71 |

Number of pages | 18 |

Journal | Theoretical Computer Science |

Volume | 393 |

Issue number | 1-3 |

DOIs | |

State | Published - Mar 20 2008 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*393*(1-3), 54-71. https://doi.org/10.1016/j.tcs.2007.10.047