### 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 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

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

}

*Theoretical Computer Science*, vol. 393, no. 1-3, pp. 54-71. https://doi.org/10.1016/j.tcs.2007.10.047

**Approximating the online set multicover problems via randomized winnowing.** / Berman, Piotr; DasGupta, Bhaskar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximating the online set multicover problems via randomized winnowing

AU - Berman, Piotr

AU - DasGupta, Bhaskar

PY - 2008/3/20

Y1 - 2008/3/20

N2 - 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 {i0, i1, ...} ⊆ V of elements are presented online in an arbitrary order. When each element ip is presented, we are also told the collection of all (at least k) sets Sip ⊆ S and their costs to which ip belongs and we need to select additional sets from Sip if necessary such that our collection of selected sets contains at leastk sets that contain the element ip. The goal is to minimize the total cost of the selected sets.11Our 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].

AB - 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 {i0, i1, ...} ⊆ V of elements are presented online in an arbitrary order. When each element ip is presented, we are also told the collection of all (at least k) sets Sip ⊆ S and their costs to which ip belongs and we need to select additional sets from Sip if necessary such that our collection of selected sets contains at leastk sets that contain the element ip. The goal is to minimize the total cost of the selected sets.11Our 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].

UR - http://www.scopus.com/inward/record.url?scp=39649102776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39649102776&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2007.10.047

DO - 10.1016/j.tcs.2007.10.047

M3 - Article

AN - SCOPUS:39649102776

VL - 393

SP - 54

EP - 71

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1-3

ER -