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Successive Linear Programs for Computing All Integral Points in a Minkowski Sum

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Advances in Informatics (PCI 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3746))

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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.

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Authors and Affiliations

  1. Dept. of Informatics and Telecommunications, National University of Athens, Greece

    Ioannis Z. Emiris & Kyriakos Zervoudakis

Authors
  1. Ioannis Z. Emiris
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  2. Kyriakos Zervoudakis
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Editors and Affiliations

  1. Department of Computer and Communication Engineering, University of Thessaly, Glavani 37, 382 21, Volos, Greece

    Panayiotis Bozanis

  2. Department of Computer and Communication Engineering, University of Thessaly, 382 21, Volos, Greece

    Elias N. Houstis

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© 2005 Springer-Verlag Berlin Heidelberg

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Emiris, I.Z., Zervoudakis, K. (2005). Successive Linear Programs for Computing All Integral Points in a Minkowski Sum. In: Bozanis, P., Houstis, E.N. (eds) Advances in Informatics. PCI 2005. Lecture Notes in Computer Science, vol 3746. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11573036_9

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  • DOI: https://doi.org/10.1007/11573036_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29673-7

  • Online ISBN: 978-3-540-32091-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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