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On the Chvátal rank of polytopes in the 0/1 cube

Identifieur interne : 000000 ( PascalFrancis/Curation ); suivant : 000001

On the Chvátal rank of polytopes in the 0/1 cube

Auteurs : A. Bockmayr [France] ; F. Eisenbrand [Allemagne] ; M. Hartmann [États-Unis] ; A. S. Schulz [États-Unis]

Source :

RBID : Pascal:00-0001592

Descripteurs français

English descriptors

Abstract

Given a polytope P ⊆ Rn, the Chvátal-Gomory procedure computes iteratively the integer hull P1 of P. The Chvátal rank of P is the minimal number of iterations needed to obtain P1. It is always finite, but already the Chvátal rank of polytopes in R2 can be arbitrarily large. In this paper, we study polytopes in the 0/1 cube, which are of particular interest in combinatorial optimization. We show that the Chvátal rank of any polytope P ⊆ [0,1]n is O(n3 log n) and prove the linear upper and lower bound n for the case P ∩ Zn = Ø.
pA  
A01 01  1    @0 0166-218X
A02 01      @0 DAMADU
A03   1    @0 Discrete appl. math.
A05       @2 98
A06       @2 1-2
A08 01  1  ENG  @1 On the Chvátal rank of polytopes in the 0/1 cube
A11 01  1    @1 BOCKMAYR (A.)
A11 02  1    @1 EISENBRAND (F.)
A11 03  1    @1 HARTMANN (M.)
A11 04  1    @1 SCHULZ (A. S.)
A14 01      @1 Université Henri Poincaré, LORIA, Campus scientifique, B.P. 239 @2 54506 Vandoeuvre-lès-Nancy @3 FRA @Z 1 aut.
A14 02      @1 Max-Planck-Institut für Informatik, Im Stadtwald @2 66123 Saarbrücken @3 DEU @Z 2 aut.
A14 03      @1 University of North Carolina at Chapel Hill, Department of Operations Research @2 Chapel Hill, NC 27599-3180 @3 USA @Z 3 aut.
A14 04      @1 MIT, Sloan School of Management and Operations Research Center, E53-361 @2 Cambridge, MA 02142 @3 USA @Z 4 aut.
A20       @1 21-27
A21       @1 1999
A23 01      @0 ENG
A43 01      @1 INIST @2 18287 @5 354000080488120020
A44       @0 0000 @1 © 2000 INIST-CNRS. All rights reserved.
A45       @0 13 ref.
A47 01  1    @0 00-0001592
A60       @1 P
A61       @0 A
A64 01  1    @0 Discrete applied mathematics
A66 01      @0 NLD
C01 01    ENG  @0 Given a polytope P ⊆ Rn, the Chvátal-Gomory procedure computes iteratively the integer hull P1 of P. The Chvátal rank of P is the minimal number of iterations needed to obtain P1. It is always finite, but already the Chvátal rank of polytopes in R2 can be arbitrarily large. In this paper, we study polytopes in the 0/1 cube, which are of particular interest in combinatorial optimization. We show that the Chvátal rank of any polytope P ⊆ [0,1]n is O(n3 log n) and prove the linear upper and lower bound n for the case P ∩ Zn = Ø.
C02 01  X    @0 001A02F02
C03 01  X  FRE  @0 Géométrie @5 01
C03 01  X  ENG  @0 Geometry @5 01
C03 01  X  SPA  @0 Geometría @5 01
C03 02  X  FRE  @0 Polytope @5 02
C03 02  X  ENG  @0 Polytope @5 02
C03 02  X  SPA  @0 Politope @5 02
C03 03  X  FRE  @0 Polyèdre @5 03
C03 03  X  ENG  @0 Polyhedron @5 03
C03 03  X  SPA  @0 Poliedro @5 03
C03 04  X  FRE  @0 Rang @5 04
C03 04  X  ENG  @0 Rank @5 04
C03 04  X  SPA  @0 Rango @5 04
C03 05  X  FRE  @0 Borne supérieure @5 05
C03 05  X  ENG  @0 Upper bound @5 05
C03 05  X  SPA  @0 Cota superior @5 05
C03 06  X  FRE  @0 Optimisation combinatoire @5 06
C03 06  X  ENG  @0 Combinatorial optimization @5 06
C03 06  X  SPA  @0 Optimización combinatoria @5 06
C03 07  X  FRE  @0 Cube 0/1 @4 CD @5 96
C03 07  X  ENG  @0 Cube 0/1 @4 CD @5 96
C03 08  X  FRE  @0 Polytope dans cube 0/1 @4 CD @5 97
C03 08  X  ENG  @0 Polytope in 01 cube @4 CD @5 97
C03 09  X  FRE  @0 Rang Chvatal @4 CD @5 98
C03 09  X  ENG  @0 Chvatal rank @4 CD @5 98
C03 10  X  FRE  @0 Envéloppe entière @4 CD @5 99
C03 10  X  ENG  @0 Integer hull @4 CD @5 99
N21       @1 003

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. It is always finite, but already the Chvátal rank of polytopes in R
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can be arbitrarily large. In this paper, we study polytopes in the 0/1 cube, which are of particular interest in combinatorial optimization. We show that the Chvátal rank of any polytope P ⊆ [0,1]
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