On the complexity of the Bernstein combinatorial problem

Dominique Michelucci, Sebti Foufou, Arnaud Kubicki

    Research output: Contribution to journalArticle

    Abstract

    Every multivariate polynomial p(x), x = (x1,...,xn) G [0, 1]n, is enclosed in the interval given by the smallest and the greatest of its coefficients in the Tensorial Bernstein Basis (TBB). Knowing that the total number of these TBB coefficients is exponential with respect to the number of variables n, IIni=1(1 + di), even if all partial degrees di equal 1, a combinatorial problem arises: is it possible to compute in polynomial time the smallest and the greatest coefficients? This article proves that the 3-SAT problem, known to be NP-complete, polynomially reduces to the above defined combinatorial problem, which let us consequently conclude that this problem is NP-hard.

    Original languageEnglish (US)
    Pages (from-to)22-33
    Number of pages12
    JournalReliable Computing
    Volume17
    StatePublished - Dec 1 2012

    Fingerprint

    Combinatorial Problems
    Bernstein Basis
    Polynomials
    Coefficient
    NP-complete problem
    Computational complexity
    Multivariate Polynomials
    Polynomial time
    Partial
    Interval

    Keywords

    • Bernstein polynomials
    • Combinatorial complexity
    • Interval arithmetics
    • Tensorial Bernstein Basis

    ASJC Scopus subject areas

    • Software
    • Computational Mathematics
    • Applied Mathematics

    Cite this

    Michelucci, D., Foufou, S., & Kubicki, A. (2012). On the complexity of the Bernstein combinatorial problem. Reliable Computing, 17, 22-33.

    On the complexity of the Bernstein combinatorial problem. / Michelucci, Dominique; Foufou, Sebti; Kubicki, Arnaud.

    In: Reliable Computing, Vol. 17, 01.12.2012, p. 22-33.

    Research output: Contribution to journalArticle

    Michelucci, D, Foufou, S & Kubicki, A 2012, 'On the complexity of the Bernstein combinatorial problem', Reliable Computing, vol. 17, pp. 22-33.
    Michelucci D, Foufou S, Kubicki A. On the complexity of the Bernstein combinatorial problem. Reliable Computing. 2012 Dec 1;17:22-33.
    Michelucci, Dominique ; Foufou, Sebti ; Kubicki, Arnaud. / On the complexity of the Bernstein combinatorial problem. In: Reliable Computing. 2012 ; Vol. 17. pp. 22-33.
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