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

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 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.

Towards the Detection of Multiple Reflections in Time-Domain Em Inverse Scattering of Multi-Layered Media

“…(2), we here recall a methodology which employs the basis reduction algorithm and guarantees a polynomial processing time [16].…”

“…(8) exists, and the numbers N 1 and N 2 are chosen large enough [16], the (n + 1) × (n − m + 1) submatrixR of reduced basis obtained after applying the basis reduction algorithm in [17] would have the form:…”

Lattice based extended formulations for integer linear equality systems

“…where N 1 and N 2 are appropriately chosen positive integers, see [3]. The first n−m+1 columns of the reduced basis B will be…”

“…Recently, several hard integer programming feasibility problems have been successfully tackled using a lattice reformulation proposed by Aardal et al [2,3]. These problems include various equality knapsacks of Cornuéjols et al [7], Aardal and Lenstra [4], market split instances of Cornuéjols and Dawande [6,1], and financial planning instances of Louveaux and Wolsey [14].…”

Basis Reduction Methods

“…Somewhat later Kannan developed an improved variant [2,3], which-to date-has the best theoretical complexity for integer programming feasibility. The goal of this survey is to review lattice-based methods to solve (IP), focusing on Lenstra's and Kannan's algorithms, which are by now considered ''classical,'' and the more recent reformulation methods of Aardal et al [4], and Krishnamoorthy and Pataki [5]. all intersect P, so branch-and-bound trying to prove infeasibility generates 10 subproblems, when branching on x 1 .…”