Theory of linear and integer programming
Theory of linear and integer programming
Minkowski addition of polytopes: computational complexity and applications to Gro¨bner bases
SIAM Journal on Discrete Mathematics
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
ICS '96 Proceedings of the 10th international conference on Supercomputing
Parallelizing compiler techniques based on linear inequalities
Parallelizing compiler techniques based on linear inequalities
Handbook of discrete and computational geometry
Multihomogeneous resultant formulae by means of complexes
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Enumerating a subset of the integer points inside a Minkowski sum
Computational Geometry: Theory and Applications
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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.