The LLL Algorithm and Integer Programming

Aardal, Karen; Eisenbrand, Friedrich

doi:10.1007/978-3-642-02295-1_9

Cited by 2 publications

(4 citation statements)

References 46 publications

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“…However, this contradicts with the well-known fact that det{B T B} is invariant for a given lattice. Similar work may be found in Ling and Mow [39], though they did not summarize their related work as clearly as we state in proposition 1 with the help of formulae (6a) to (8). Nevertheless, we should note that proposition 1 is still slightly different from the work of Ling and Mow [39] in two senses: (a) while Ling and Mow [39] directly implemented the sorted QR technique to re-arrange the basis vectors, we do not assume any sorting in proposition 1 and (b) as a result of (a), the proofs given here and in Ling and Mow [39] are essentially different.…”

Section: Propositionmentioning

confidence: 55%

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Experimental quality evaluation of lattice basis reduction methods for decorrelating low-dimensional integer least squares problems

2013

EURASIP J. Adv. Signal Process.

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Reduction can be important to aid quickly attaining the integer least squares (ILS) estimate from noisy data. We present an improved LLL algorithm with fixed complexity by extending a parallel reduction method for positive definite quadratic forms to lattice vectors. We propose the minimum angle of a reduced basis as an alternative quality measure of orthogonality, which is intuitively more appealing to measure the extent of orthogonality of a reduced basis. Although the LLL algorithm and its variants have been widely used in practice, experimental simulations were only carried out recently and limited to the quality measures of the Hermite factor, practical running behaviors and reduced Gram-Schmidt coefficients. We conduct a large scale of experiments to comprehensively evaluate and compare five reduction methods for decorrelating ILS problems, including the LLL algorithm, its variant with deep insertions and our improved LLL algorithm with fixed complexity, based on six quality measures of reduction. We use the results of the experiments to investigate the mean running behaviors of the LLL algorithm and its variants with deep insertions and the sorted QR ordering, respectively. The improved LLL algorithm with fixed complexity is shown to perform as well as the LLL algorithm with deep insertions with respect to the quality measures on length reduction but significantly better than this LLL variant with respect to the other quality measures. In particular, our algorithm is of fixed complexity, but the LLL algorithm with deep insertions could seemingly not be terminated in polynomial time of the dimension of an ILS problem. It is shown to perform much better than the other three reduction methods with respect to all the six quality measures. More than six millions of the reduced Gram-Schmidt coefficients from each of the five reduction methods clearly show that they are not uniformly distributed but depend on the reduction algorithms used. The simulation results of the reduced Gram-Schmidt coefficients have clearly shown that our improved LLL algorithm tends to produce small reduced Gram-Schmidt coefficients near zero with a larger probability and large reduced Gram-Schmidt coefficients near both ends of 0.5 and −0.5 with a smaller probability.

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Section: Propositionmentioning

confidence: 55%

“…. , b m in R n , the reduction by repeating the process from (6a) to (8) always converges in a finite number of iterations.…”

Section: Propositionmentioning

confidence: 99%

Experimental quality evaluation of lattice basis reduction methods for decorrelating low-dimensional integer least squares problems

2013

EURASIP J. Adv. Signal Process.

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“…Since then a number of lattice-based approaches have been proposed for integer linear programming problems, see e.g. the surveys [3,1] and the recent papers [26,4,34,31,2] among many others. Recent research has also considered convex quadratic objectives (as is the case for the ILS problem), see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…2 Distance to Integrality and Size of the BB Tree Let x R denote the optimal solution to the RLS problem (2) and x I the solution to the ILS problem (1). We define the distance to integrality as the Euclidean distance between the real and integer optimal solutions to the ILS problem:…”

Section: Introductionmentioning

confidence: 99%

Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

Anjos

Chang

2014

J Glob Optim

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The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proportional to this distance. Since we cannot know the distance to integrality a priori, we prove that the norm of the Moore-Penrose generalized inverse of the matrix of coefficients is a key factor for bounding this distance, and then we propose a preconditioning method to reduce this norm using lattice reduction techniques. We also propose a set of valid box constraints that help accelerate the RRBB method. Our computational results show that the proposed preconditioning significantly reduces the size of the RRBB enumeration tree, that the preconditioning combined with the proposed set of box constraints can significantly reduce the computational time of RRBB, and that the resulting RRBB method can outperform the Schnorr and Eucher method, a widely used method for solving integer least squares problems, on some types of problem data.

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The LLL Algorithm and Integer Programming

Cited by 2 publications

References 46 publications

Experimental quality evaluation of lattice basis reduction methods for decorrelating low-dimensional integer least squares problems

Experimental quality evaluation of lattice basis reduction methods for decorrelating low-dimensional integer least squares problems

Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

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