On the Structure of Reduced Kernel Lattice Bases

Abstract

Lattice-based reformulation techniques have been used successfully both theoretically and computationally. One such reformulation is obtained from the kernel lattice associated with an input matrix. Some of the hard instances in the literature that have been successfully tackled by lattice-based techniques have randomly generated input. Since the considered instances are very hard even in low dimension, less experience is available for larger instances. Recently, we have studied larger instances and observed that the LLL-reduced basis of the kernel lattice has a specific sparse structure. In particular, this translates into a map in which some of the original variables get a “rich”' translation into a new variable space, whereas some variables are only substituted in the new space. If an original variable is important in the sense of branching or cutting planes, this variable should be translated in a nontrivial way. In this paper we partially explain, through a probabilistic analysis, the obtained structure of the LLL-reduced basis in the case that the input matrix consists of one row. The key ingredient is a bound on the probability that the LLL-algorithm will interchange two subsequent basis vectors.

Tackling A Class of Hard Subset-Sum Problems: Integration of Lattice Attacks with Disaggregation Techniques

“…with the notation ⌊•⌋ denoting the floor truncate integer of a rational number, for instance, ⌊2 2 3 ⌋ = 2 and ⌊−2 2 3 ⌋ = −3. It is not hard to observe that 0 ≤ d < M , 0 ≤ w < b, and 0 ≤ c i < M , 0 ≤ v i < a i for i = 1, 2, .…”

“…Now it is ready to present our observations and basic theorems regarding NJPs. Consider coefficients of the disaggregated equations v (1) x+k (1) = w (1) and v (2) x+k (2) = w (2) , our observations are summarized in Observation 2 and Observation 3. (c) For 1 ≤ i = h ≤ n, ∆v i = 1 if and only if j a h = ĵ a i for some ĵ ∈ {1, 2, .…”

“…(b) After disaggregation, whether set 3 |r can cut-off the non-binary integer solution x with small Euclidean length, depends on the structure of the corresponding new kernel lattice ker Z (a T , v T ) T ). Note that, the concept of kernel lattice has been defined in item (a) of Theorem 2, which has also been studied in [2].…”

Computing a Lattice Basis Revisited

“…Some applications (such as integer programming [AvH14], linear Diophantine equations [Ili89b] and matrix gcds [SL95]) require a unimodular transform corresponding to the output basis. Alg.…”