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A Geometric Buchberger Algorithm for Integer Programming

A Geometric Buchberger Algorithm for Integer Programming Let IPA, c denote the family of integer programs of the form Min cx: Ax b, x Nn obtained by varying the right-hand side vector b but keeping A and c fixed. A test set for IPA, c is a set of vectors in Zn such that for each nonoptimal solution to a program in this family, there is at least one element g in this set such that g has an improved cost value as compared to . We describe a unique minimal test set for this family called the reduced Grbner basis of IPA, c. An algorithm for its construction is presented which we call a geometric Buchberger algorithm for integer programming and we show how an integer program may be solved using this test set. The reduced Grbner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A Z(n2)n of full row rank. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics of Operations Research INFORMS

A Geometric Buchberger Algorithm for Integer Programming

Mathematics of Operations Research , Volume 20 (4): 21 – Nov 1, 1995
21 pages

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References (16)

Publisher
INFORMS
Copyright
Copyright © INFORMS
Subject
Research Article
ISSN
0364-765X
eISSN
1526-5471
DOI
10.1287/moor.20.4.864
Publisher site
See Article on Publisher Site

Abstract

Let IPA, c denote the family of integer programs of the form Min cx: Ax b, x Nn obtained by varying the right-hand side vector b but keeping A and c fixed. A test set for IPA, c is a set of vectors in Zn such that for each nonoptimal solution to a program in this family, there is at least one element g in this set such that g has an improved cost value as compared to . We describe a unique minimal test set for this family called the reduced Grbner basis of IPA, c. An algorithm for its construction is presented which we call a geometric Buchberger algorithm for integer programming and we show how an integer program may be solved using this test set. The reduced Grbner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A Z(n2)n of full row rank.

Journal

Mathematics of Operations ResearchINFORMS

Published: Nov 1, 1995

Keywords: Keywords : integer programming ; test sets ; reduced Gröbner ; Buchberger algorithm ; fiber ; Hilbert bases

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