1 5 Ju l 2 00 9 Basis Reduction , and the Complexity of Branch - and - Bound Gábor Pataki and Mustafa Tural

Basis Reduction and the Complexity of Brand-and-Bound

Basis Reduction Methods

where P is a polyhedron described by inequalities. The work of Lenstra in 1979 [1] answered one of the most challenging questions in the theory of integer programming by presenting an algorithm to solve (IP), whose running time is polynomial, when n, the dimension is fixed. This paper pioneered the use of lattice basis reduction in integer programming, and initiated an interest in polynomial re...

On sublattice determinants in reduced bases

Lenstra, Lenstra, and Lovász in [7] proved several inequalities showing that the vectors in an LLL-reduced basis are short, and near orthogonal. Here we present generalizations, from which with k = 1, and k = n we can recover their inequalities: Theorem 1. Let b1, . . . , bn ∈ R be an LLL-reduced basis of the lattice L, and d1, . . . , dk arbitrary linearly independent vectors in L. Then ‖b1 ‖ ...

Parallel Approximation, and Integer Programming Reformulation

We show that in a knapsack feasibility problem an integral vector p, which is short, and near parallel to the constraint vector gives a branching direction with small integer width. We use this result to analyze two computationally efficient reformulation techniques on low density knapsack problems. Both reformulations have a constraint matrix with columns reduced in the sense of Lenstra, Lenst...

Unifying LLL inequalities

The Lenstra, Lenstra, and Lovász (abbreviated as LLL) basis reduction algorithm computes a basis of a lattice consisting of short, and near orthogonal vectors. The quality of an LLL reduced basis is expressed by three fundamental inequalities, and it is natural to ask, whether these have a common generalization. In this note we find unifying inequalities. Our main result is Theorem 1. Let b1, ....