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An Efficient Parametric Linear Programming Solver and Application to Polyhedral Projection

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Static Analysis (SAS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11822))

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Abstract

Polyhedral projection is a main operation of the polyhedron abstract domain. It can be computed via parametric linear programming (PLP), which is more efficient than the classic Fourier-Motzkin elimination method.

In prior work, PLP was done in arbitrary precision rational arithmetic. In this paper, we present an approach where most of the computation is performed in floating-point arithmetic, then exact rational results are reconstructed.

We also propose a workaround for a difficulty that plagued previous attempts at using PLP for computations on polyhedra: in general the linear programming problems are degenerate, resulting in redundant computations and geometric descriptions.

Grenoble INP—Institute of Engineering Univ. Grenoble Alpes

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Notes

  1. 1.

    Now distributed as part of APRON http://apron.cri.ensmp.fr/library/.

  2. 2.

    This is the canonical form of the LP problem. All the LP problems can be transformed into this form.

  3. 3.

    The GNU Linear Programming Toolkit (GLPK) is a linear programming solver implemented in floating-point arithmetic. https://www.gnu.org/software/glpk/.

  4. 4.

    There also exist parametric linear programs where the parameters are in the constant terms of the inequalities, we do not consider them here.

  5. 5.

    We use Flint, which provides exact rational scalar, vector and matrix computations, including solving of linear systems. http://www.flintlib.org/.

  6. 6.

    There may be rows of all zeros in the bottom of the matrix.

  7. 7.

    https://github.com/antoinemine/apron.

  8. 8.

    The minimization is still computed in floating-point numbers.

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Authors and Affiliations

  1. Univ. Grenoble Alpes, CNRS, Grenoble INP, 38000, Grenoble, France

    Hang Yu & David Monniaux

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  1. Hang Yu
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  2. David Monniaux
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Correspondence to Hang Yu .

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  1. University of Colorado, Boulder, CO, USA

    Bor-Yuh Evan Chang

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Yu, H., Monniaux, D. (2019). An Efficient Parametric Linear Programming Solver and Application to Polyhedral Projection. In: Chang, BY. (eds) Static Analysis. SAS 2019. Lecture Notes in Computer Science(), vol 11822. Springer, Cham. https://doi.org/10.1007/978-3-030-32304-2_11

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  • DOI: https://doi.org/10.1007/978-3-030-32304-2_11

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  • Print ISBN: 978-3-030-32303-5

  • Online ISBN: 978-3-030-32304-2

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