Hard equality constrained integer knapsacks

K.I. Aardal, A.K. Lenstra

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Abstract

We consider the following integer feasibility problem: "Given positive integer numbers a 0, a 1,..., a n, with gcd(a 1,..., a n) = 1 and a = (a 1,..., a n), does there exist a nonnegative integer vector x satisfying ax = a 0?" Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as small as ten. We observe that not only the sizes of the numbers a 0, a 1,..., a n, but also their structure, have a large impact on the difficulty of the instances. Moreover, we demonstrate that the characteristics that make the instances so difficult to solve by branch-and-bound make the solution of a certain reformulation of the problem almost trivial. We accompany our results by a small computational study.
Original languageEnglish
Title of host publicationProceedings 9th IPCO (Cambridge MA, USA, May 27-29, 2002)
EditorsW.J. Cook, A.S. Schulz
Place of PublicationBerlin
PublisherSpringer
Pages350-366
ISBN (Print)3-540-43676-6
DOIs
Publication statusPublished - 2002

Publication series

NameLecture Notes in Computer Science
Volume2337
ISSN (Print)0302-9743

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    Aardal, K. I., & Lenstra, A. K. (2002). Hard equality constrained integer knapsacks. In W. J. Cook, & A. S. Schulz (Eds.), Proceedings 9th IPCO (Cambridge MA, USA, May 27-29, 2002) (pp. 350-366). (Lecture Notes in Computer Science; Vol. 2337). Springer. https://doi.org/10.1007/3-540-47867-1_25