### Abstract

Every multivariate polynomial p(x), x = (x_{1},...,x_{n}) 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, II^{n}_{i}=1(1 + d_{i}), 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 language | English (US) |
---|---|

Pages (from-to) | 22-33 |

Number of pages | 12 |

Journal | Reliable Computing |

Volume | 17 |

State | Published - Dec 1 2012 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software
- Computational Mathematics
- Applied Mathematics

### Cite this

*Reliable Computing*,

*17*, 22-33.

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

Research output: Contribution to journal › Article

*Reliable Computing*, vol. 17, pp. 22-33.

}

TY - JOUR

T1 - On the complexity of the Bernstein combinatorial problem

AU - Michelucci, Dominique

AU - Foufou, Sebti

AU - Kubicki, Arnaud

PY - 2012/12/1

Y1 - 2012/12/1

N2 - 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.

AB - 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.

KW - Bernstein polynomials

KW - Combinatorial complexity

KW - Interval arithmetics

KW - Tensorial Bernstein Basis

UR - http://www.scopus.com/inward/record.url?scp=84872711963&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872711963&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 22

EP - 33

JO - Reliable Computing

JF - Reliable Computing

SN - 1385-3139

ER -