Shrinking integer lattices

Integer Optimization and Lattices

• Lattices. We will see basic properties of lattices, followed by Minkowski’s Theorem which guarantees that any symmetric convex set with volume bigger than 2 must have an non-zero integer point. We will show an application of Minkowski’s theorem to Diophantine approximations. Then we will introduce the powerful concept of Lattice basis reduction which finds an almost orthogonal basis for a lat...

Camera Placement in Integer Lattices

The camera placement problem concerns the placement of a xed number of point-cameras on the integer lattice of d-tuples of integers in order to maximize their visibility. We give a caracterization of optimal conng-urations of size s less than 5 d and use it to compute in time O(s log s) an optimal abstract connguration under the assumption that the visibility of a connguration is computable in ...

BOOK OF ABSTRACTS Isoperimetry in integer lattices

The edge isoperimetric problem for a graph G is to determine, for each n, the minimum number of edges leaving any set of n vertices. Exact solutions are known only in special cases, for example when G is the usual integer lattice. The most natural open problem was to answer this question for the ‘strong lattice’, with edges between points at l∞ distance 1. Whilst studying this question we in fa...

Completely Empty Pyramids on Integer Lattices

A Note on Integer Factorization Using Lattices

We revisit Schnorr’s lattice-based integer factorization algorithm, now with an effective point of view. We present effective versions of Theorem 2 of [11], as well as new properties of the Prime Number Lattice bases of Schnorr and Adleman.