Enumerating a subset of the integer points inside a Minkowski sum

Authors:
Ioannis Z. Emiris
Affiliations:
INRIA, B.P. 93, Sophia-Antipolis 06902, France
Venue:
Computational Geometry: Theory and Applications
Year:
2002

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Abstract

Sparse elimination exploits the structure of algebraic equations in order to obtain tighter bounds on the number of roots and better complexity in numerically approximating them. The model of sparsity is of combinatorial nature, thus leading to certain problems in general-dimensional convex geometry. This work addresses one such problem, namely the computation of a certain subset of integer points in the interior of integer convex polytopes. These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm. Complexity bounds for our algorithm are derived under certain hypotheses, in terms of output-size and the sparsity parameters. A public domain implementation is described and its performance studied. Linear optimization lies at the inner loop of the algorithm, hence we analyze the structure of the linear programs and compare different implementations.