computational complexity

, Volume 10, Issue 2, pp 139–154

Complexity of Positivstellensatz proofs for the knapsack

  • D. Grigoriev

DOI: 10.1007/s00037-001-8192-0

Cite this article as:
Grigoriev, D. Comput. complex. (2001) 10: 139. doi:10.1007/s00037-001-8192-0


A lower bound is established on degrees of Positivstellensatz calculus refutations (over a real field) introduced in (Grigoriev & Vorobjov 2001; Grigoriev 2001) for the knapsack problem. The bound depends on the values of coefficients of an instance of the knapsack problem: for certain values the lower bound is linear and for certain values the upper bound is constant, while in the polynomial calculus the degree is always linear (regardless of the values of coefficients) (Impagliazzo et al. 1999). This shows that the Positivstellensatz calculus can be strictly stronger than the polynomial calculus from the point of view of the complexity of the proofs.

Keywords. Polynomial calculus, Positivstellensatz proofs, complexity of the knapsack.

Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • D. Grigoriev
    • 1
  1. 1.IRM Université de Rennes-1, Campus de Beaulieu, 35042 Rennes, France, e-mail:, http://www.maths.univ-rennes1.frFR