Successive linear programs for computing all integral points in a minkowski sum

Abstract

The computation of all integral points in Minkowski (or vector) sums of convex lattice polytopes of arbitrary dimension appears as a subproblem in algebraic variable elimination, parallel compiler code optimization, polyhedral combinatorics and multivariate polynomial multiplication. We use an existing approach that avoids the costly construction of the Minkowski sum by an incremental process of solving Linear Programming (LP) problems. Our main contribution is to exploit the similarities between LP problems in the tree of LP instances, using duality theory and the two-phase simplex algorithm. Our public domain implementation improves substantially upon the performance of the above mentioned approach and is faster than porta on certain input families; besides, the latter requires a description of the Minkowski sum which has high complexity. Memory consumption limits commercial or free software packages implementing multivariate polynomial multiplication, whereas ours can solve all examined data, namely of dimension up to 9, using less than 2.7 MB (before actually outputting the points) for instances yielding more than 3 million points. © Springer-Verlag Berlin Heidelberg 2005.