Algorithms for the Densest Sub-Lattice Problem

We give algorithms for computing the densest k-dimensional sublattice of an arbitrary lattice, and related problems. This is an important problem in the algorithmic geometry of numbers that includes as special cases Rankin’s problem (which corresponds to the densest sublattice problem with respect to the Euclidean norm, and has applications to the design of lattice reduction algorithms), and the shortest vector problem for arbitrary norms (which corresponds to setting k = 1) and its dual (k = n− 1). Remarkably, our algorithm works for any norm and has running time kO(k·n). In particular, the algorithm runs in single exponential time 2 for any constant k = O(1). ∗Georgia Tech. Atlanta, GA, USA. Email: [email protected] †University of California, San Diego. 9500 Gilman Dr., Mail Code 0404, La Jolla, CA 92093, USA. Email: [email protected]