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Summary: Solving a system of linear diophantine equations with
lower and upper bounds on the variables
Karen Aardal Cor A. J. Hurkens y Arjen K. Lenstra z
Abstract
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 rst nds a short vector satisfying the system of diophantine equations, and a set of
vectors belonging to the null-space 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 satis es 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.
AMS 2000 Subject classi cation: Primary: 90C10. Secondary: 45F05, 11Y50.
OR/MS subject classi cation: Programming, Integer, Algorithms, Relaxation.
Key words: Lattice basis reduction, short vectors, linear diophantine equations.
1 Introduction and problem description
We develop an algorithm for solving the following integer feasibility problem:
Does there exist a vector x 2 ZZ n such that Ax = d; 0 x u? (1)
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