On the smallest ratio problem of lattice bases

Let(b1, . . .,bn) be a lattice basis with Gram-Schmidt orthogonalization(b1, . . . ,b∗n), the quantities‖b1‖/‖bi ‖for i = 1, . . . , n play important roles in analyzing lattice reduction algorithms and lattice enumeration algorithms.In this paper, we study the problem of minimizing the quantity‖b1‖/‖bn‖ over all bases(b1, . . .,bn) of a givenn-dimensional lattice. We first prove that there exists a basis(b1, . . .,bn) for any lattice L of dimension n suchthat‖b1‖ =minv∈L\{0} ‖v‖,‖b1‖/‖bi ‖ ≤ i and‖bi‖/‖bi ‖ ≤ i1.5 for 1 ≤ i ≤ n. This leads us to introduce a newNP-hard computational problem, that is, the smallest ratio problem (SRP): given an n-dimensional lattice L, finda basis(b1, . . .,bn) of L such that‖b1‖/‖bn‖ is minimal. The problem inspires the new lattice invariant μn(L) =min{‖b1‖/‖bn‖ :(b1, . . .,bn) is a basis of L} and new lattice constant μn = max μn(L) over all n-dimensionallattices L: both the minimum and maximum are justified. The properties of μn(L) and μn are discussed. We alsopresent an exact algorithm and an approximation algorithm for SRP.To the best of our knowledge, this is the first sound study of SRP. Our work provides a new perspective onboth the quality limits of lattice reduction algorithms and complexity estimates of enumeration algorithms.