Some lower bounds for the complexity of the linear programming feasibility problem over the reals
We analyze the arithmetic complexity of the linear programming feasibility problem over the reals. For the case of polyhedra defined by 2 n half-spaces in Rn we prove that the set I(2 n, n), of parameters describing nonempty polyhedra, has an exponential number of limiting hypersurfaces. From this g...
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2009
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v25_n1_p25_Grimson http://hdl.handle.net/20.500.12110/paper_0885064X_v25_n1_p25_Grimson |
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paper:paper_0885064X_v25_n1_p25_Grimson2025-07-30T18:23:01Z Some lower bounds for the complexity of the linear programming feasibility problem over the reals Algebraic complexity Elimination theory Limiting hypersurface Linear programming Lower bounds Algebra Geometry Linear programming Algebraic complexity Algebraic computations Arithmetic complexity Elimination theory Feasibility problem Limiting hypersurface Lower bounds Sparse representation C (programming language) We analyze the arithmetic complexity of the linear programming feasibility problem over the reals. For the case of polyhedra defined by 2 n half-spaces in Rn 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 c > 1 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 Ω (cn). Other related complexity results are stated; in particular, a lower bound for algebraic computation trees based on the notion of limiting hypersurface is presented. © 2008 Elsevier Inc. All rights reserved. 2009 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v25_n1_p25_Grimson http://hdl.handle.net/20.500.12110/paper_0885064X_v25_n1_p25_Grimson |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Algebraic complexity Elimination theory Limiting hypersurface Linear programming Lower bounds Algebra Geometry Linear programming Algebraic complexity Algebraic computations Arithmetic complexity Elimination theory Feasibility problem Limiting hypersurface Lower bounds Sparse representation C (programming language) |
| spellingShingle |
Algebraic complexity Elimination theory Limiting hypersurface Linear programming Lower bounds Algebra Geometry Linear programming Algebraic complexity Algebraic computations Arithmetic complexity Elimination theory Feasibility problem Limiting hypersurface Lower bounds Sparse representation C (programming language) Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| topic_facet |
Algebraic complexity Elimination theory Limiting hypersurface Linear programming Lower bounds Algebra Geometry Linear programming Algebraic complexity Algebraic computations Arithmetic complexity Elimination theory Feasibility problem Limiting hypersurface Lower bounds Sparse representation C (programming language) |
| description |
We analyze the arithmetic complexity of the linear programming feasibility problem over the reals. For the case of polyhedra defined by 2 n half-spaces in Rn 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 c > 1 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 Ω (cn). Other related complexity results are stated; in particular, a lower bound for algebraic computation trees based on the notion of limiting hypersurface is presented. © 2008 Elsevier Inc. All rights reserved. |
| title |
Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| title_short |
Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| title_full |
Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| title_fullStr |
Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| title_full_unstemmed |
Some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| title_sort |
some lower bounds for the complexity of the linear programming feasibility problem over the reals |
| publishDate |
2009 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v25_n1_p25_Grimson http://hdl.handle.net/20.500.12110/paper_0885064X_v25_n1_p25_Grimson |
| _version_ |
1840323349134180352 |