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

Piotr Berman, Bhaskar DasGupta, Eduardo Sontag

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    10 Citations (Scopus)

    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 e2 ≈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 languageEnglish (US)
    Pages (from-to)39-50
    Number of pages12
    JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume3122
    StatePublished - Dec 1 2004

    Fingerprint

    Gene Networks
    Reverse engineering
    Reverse Engineering
    Approximation algorithms
    Randomized Algorithms
    Approximation Algorithms
    Genes
    Proteins
    Protein
    Computational complexity
    Polynomials
    Randomized Rounding
    Increasing Functions
    Combinatorial Problems
    Approximation
    Greedy Algorithm
    Polynomial-time Algorithm
    Cascade
    Computational Complexity
    Coverage

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

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    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 e2 ≈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.",
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    N2 - 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 e2 ≈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.

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