### Abstract

In this paper we investigate the computational complexities of a combinatorial problem that arises in the reverse engineering of protein and gene networks. Our contributions are as follows: - We abstract a combinatorial version of the problem and observe that this is "equivalent" to the set multicover problem when the "coverage" factor k is a function of the number of elements n of the universe. An important special case for our application is the case in which k = n-1. - We observe that the standard greedy algorithm produces an approximation ratio of Ω(log n) even if k is "large" i.e. k = n-c for some constant c ≥ 0. - Let 1 ≤ a ≤ n denotes the maximum number of elements in any given set in our set multicover problem. Then, we show that a non-trivial analysis of a simple randomized polynomial-time approximation algorithm for this problem yields an expected approximation ratio E[r(a, k)] that is an increasing function of a/k. The behavior of E[r(a,k)] is "roughly" as follows: it is about ln(a/k) when a/k is at least about e^{2} ≈7.39, and for smaller values of a/k it decreases towards 2 exponentially with increasing k with lim _{a/k→0} E[r(a, k)] ≤ 2. Our randomized algorithm is a cascade of a deterministic and a randomized rounding step parameterized by a quantity β followed by a greedy solution for the remaining problem.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Editors | Klaus Jansen, Sanjeev Khanna, Jose D. P. Rolim, Dana Ron |

Publisher | Springer Verlag |

Pages | 39-50 |

Number of pages | 12 |

ISBN (Print) | 3540228942, 9783540228943 |

DOIs | |

State | Published - 2004 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3122 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(pp. 39-50). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3122). Springer Verlag. https://doi.org/10.1007/978-3-540-27821-4_4