Some lower bounds for the complexity of the linear programming feasibility problem over the reals

Authors:
Rafael Grimson;Bart Kuijpers
Affiliations:
Theoretical Computer Science Group, Hasselt University and Transnational University of Limburg, Belgium and Departamento de Matemática, Universidad de Buenos Aires, Argentina;Theoretical Computer Science Group, Hasselt University and Transnational University of Limburg, Belgium
Venue:
Journal of Complexity
Year:
2009

Citing 14
Cited 0

The complexity of linear problems in fields

Journal of Symbolic Computation
Real quantifier elimination is doubly exponential

Journal of Symbolic Computation
Solving systems of polynomial inequalities in subexponential time

Journal of Symbolic Computation
Quantifier elimination: Optimal solution for two classical examples

Journal of Symbolic Computation
On the computational complexity and geometry of the first-order theory of the reals. Par I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals

Journal of Symbolic Computation
On the intrinsic complexity of elimination theory

Journal of Complexity - Special issue: invited articles dedicated to J. F. Traub on the occasion of his 60th birthday
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition

Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Ein Entscheidungsverfahren für die Theorie der reell- abgeschlossenen Körper

Komplexität von Entscheidungsproblemen, Ein Seminar
The complexity of elementary algebra and geometry

STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Lower bounds for algebraic computation trees

STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)

Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Factoring bivariate sparse (lacunary) polynomials

Journal of Complexity
The complexity of quantifier elimination and cylindrical algebraic decomposition

Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the combinatorial and algebraic complexity of quantifier elimination

SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science

Quantified Score

Hi-index	0.00

Visualization

Abstract

We analyze the arithmetic complexity of the linear programming feasibility problem over the reals. For the case of polyhedra defined by 2n half-spaces in R^n we prove that the set I^(^2^n^,^n^), of parameters describing nonempty polyhedra, has an exponential number of limiting hypersurfaces. From this geometric result we obtain, as a corollary, the existence of a constant c1 such that, if dense or sparse representation is used to code polynomials, the length of any quantifier-free formula expressing the set I^(^2^n^,^n^) is bounded from below by @W(c^n). Other related complexity results are stated; in particular, a lower bound for algebraic computation trees based on the notion of limiting hypersurface is presented.