Abstract
Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems. The two main computational goals are (i) finding a system of generators of the set of lattice points and (ii) counting elements degree-wise in a generating function, the Hilbert Series. In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid.
We will present some improvements to the Normaliz algorithm by subdividing simplicial cones with huge volumes. In the first approach the subdivision points are found by integer programming techniques. For this purpose we interface to the integer programming solver SCIP to our software. In the second approach we try to find good subdivision points in an approximating overcone that is faster to compute.
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References
Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–41 (2009). http://mpc.zib.de/index.php/MPC/article/view/4
Bruns, W., Gubeladze, J.: Polytopes, Rings and K-Theory. Springer, New York (2009)
Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids. http://www.math.uos.de/normaliz
Bruns, W., Ichim, B.: Normaliz: algorithms for affine monoids and rational cones. J. Algebra 324, 1098–1113 (2010)
Bruns, W., Ichim, B., Sger, C.: The power of pyramid decompositions in Normaliz. J. Symb. Comp. 74, 513–536 (2016)
Pottier, L.: The Euclide algorithm in dimension n. Research report, ISSAC 1996. ACM Press (1996)
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© 2016 Springer International Publishing Switzerland
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Bruns, W., Sieg, R., Söger, C. (2016). The Subdivision of Large Simplicial Cones in Normaliz. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_13
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DOI: https://doi.org/10.1007/978-3-319-42432-3_13
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