Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables

We develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction. It first finds a short vector satisfying the system of diophantine equations, and a set of vectors belonging to the nullspace of the...

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Bibliographic Details
Published inMathematics of operations research Vol. 25; no. 3; pp. 427 - 442
Main Authors Aardal, Karen, Hurkens, Cor A. J, Lenstra, Arjen K
Format Journal Article
LanguageEnglish
Published Linthicum INFORMS 01.08.2000
Institute for Operations Research and the Management Sciences
Subjects
Algorithms
Computer programming
Diophantine equation
Equations
Integer programming
Integers
Lattice basis reduction
linear diophantine equations
Linear programming
Management science
Mathematical lattices
Mathematical vectors
Matrices
Matrix
Operations research
Polynomials
Polytopes
short vectors
Studies
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Summary:We develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction. It first finds a short vector satisfying the system of diophantine equations, and a set of vectors belonging to the nullspace of the constraint matrix. Due to basis reduction, all these vectors are relatively short. The next step is to branch on linear combinations of the null-space vectors, which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists. The research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing. Our algorithm is tested with good results on real-life data, and on instances from the literature.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.25.3.427.12219