Cutting Planes and the Elementary Closure in Fixed Dimension

The elementary closureP' of a polyhedronP is the intersection ofP with all its Gomory-Chvatal cutting planes.P' is a rational polyhedron provided thatP is rational. The known bounds for the number of inequalities definingP' are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. IfP is a simplicial cone, we construct a polytopeQ, whose integral elements correspond to cutting planes ofP. The vertices of the integer hullQ Iinclude the facets ofP'. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. (1992). Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices ofQ I .

[1]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[2]  James Lee Hafner,et al.  Asymptotically fast triangulation of matrices over rings , 1991, SODA '90.

[3]  G. Nemhauser,et al.  Integer Programming , 2020 .

[4]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[5]  M. Hung,et al.  An application of the Hermite normal form in integer programming , 1990 .

[6]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[7]  William J. Cook,et al.  On integer points in polyhedra , 1992, Comb..

[8]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[9]  Friedrich Eisenbrand,et al.  Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube , 1999, IPCO.

[10]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[11]  W. Pulleyblank,et al.  Total Dual Integrality and Integer Polyhedra , 1979 .

[12]  John A. Howell,et al.  Spans in the module (Zm ) s , 1986 .

[13]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[14]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[15]  Alexander Schrijver,et al.  On Cutting Planes , 1980 .

[16]  László Lovász,et al.  The Generalized Basis Reduction Algorithm , 1990, Math. Oper. Res..

[17]  Arne Storjohann,et al.  Fast Algorithms for for Linear Algebra Modulo N , 1998, ESA.

[18]  William J. Cook,et al.  On the complexity of cutting-plane proofs , 1987, Discret. Appl. Math..

[19]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[20]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[21]  R E Gomory,et al.  Faces of an integer polyhedron. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[22]  William J. Cook.  Cutting-plane proofs in polynomial space , 1990, Math. Program..

[23]  László Lovász,et al.  On integer points in polyhedra: A lower bound , 1992, Comb..

[24]  George Labahn,et al.  Asymptotically fast computation of Hermite normal forms of integer matrices , 1996, ISSAC '96.

[25]  Matteo Fischetti,et al.  On the separation of maximally violated mod-k cuts , 1999, Math. Program..

[26]  Friedrich Eisenbrand,et al.  NOTE – On the Membership Problem for the Elementary Closure of a Polyhedron , 1999, Comb..

[27]  Friedrich Eisenbrand,et al.  On the Chvátal Rank of Polytopes in the 0/1 Cube , 1999, Discret. Appl. Math..

[28]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[29]  David G. Larman,et al.  The vertices of the knapsack polytope , 1983, Discret. Appl. Math..

[30]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[31]  G. Italiano,et al.  Algorit[h]ms - ESA '98 : 6th Annual European Symposium, Venice, Italy, August 24-26, 1998 : proceedings , 1998 .