### Abstract

In this paper we investigate the computational complexity 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 denote 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 1 as a linear function of sqrt(a / k) with lim_{a / k → 0} E [r (a, k)] = 1. 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. We also comment about the impossibility of a significantly faster convergence of E [r (a, k)] towards 1 for any polynomial-time approximation algorithm.

Original language | English (US) |
---|---|

Pages (from-to) | 733-749 |

Number of pages | 17 |

Journal | Discrete Applied Mathematics |

Volume | 155 |

Issue number | 6-7 |

DOIs | |

State | Published - Apr 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*155*(6-7), 733-749. https://doi.org/10.1016/j.dam.2004.11.009

}

*Discrete Applied Mathematics*, vol. 155, no. 6-7, pp. 733-749. https://doi.org/10.1016/j.dam.2004.11.009

**Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks.** / Berman, Piotr; DasGupta, Bhaskar; Sontag, Eduardo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks

AU - Berman, Piotr

AU - DasGupta, Bhaskar

AU - Sontag, Eduardo

PY - 2007/4/1

Y1 - 2007/4/1

N2 - In this paper we investigate the computational complexity 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 denote 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 e2 ≈ 7.39, and for smaller values of a / k it decreases towards 1 as a linear function of sqrt(a / k) with lima / k → 0 E [r (a, k)] = 1. 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. We also comment about the impossibility of a significantly faster convergence of E [r (a, k)] towards 1 for any polynomial-time approximation algorithm.

AB - In this paper we investigate the computational complexity 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 denote 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 e2 ≈ 7.39, and for smaller values of a / k it decreases towards 1 as a linear function of sqrt(a / k) with lima / k → 0 E [r (a, k)] = 1. 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. We also comment about the impossibility of a significantly faster convergence of E [r (a, k)] towards 1 for any polynomial-time approximation algorithm.

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U2 - 10.1016/j.dam.2004.11.009

DO - 10.1016/j.dam.2004.11.009

M3 - Article

VL - 155

SP - 733

EP - 749

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 6-7

ER -