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Solving a Linear Diophantine Equation with Lower and Upper Bounds on the Variables

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Integer Programming and Combinatorial Optimization (IPCO 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1412))

Abstract

We develop an algorithm for solving a linear diophantine equation with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction, and first finds short vectors satisfying the diophantine equation. The next step is to branch on linear combi- nations of these vectors, which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists. The research was motivated by the need for solving constrained linear dio- phantine equations as subproblems when designing integrated circuits for video signal processing. Our algorithm is tested with good result on real-life data.

Research partially supported by ESPRIT Long Term Research Project No. 20244 (Project ALCOM: Algorithms and Complexity in Information Technology), and by NSF grant CCR-9307391 through David B. Shmoys, Cornell University.

Research partially supported by ESPRIT Long Term Research Project No. 20244 (Project ALCOM: Algorithms and Complexity in Information Technology).

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References

  1. K. Aardal, A. K. Lenstra, and C. A. J. Hurkens. An algorithm for solving a diophantine equation with upper and lower bounds on the variables. Report UU-CS-97-40, Department of Computer Science, Utrecht University, 1997. ftp://ftp.cs.ruu.nl/pub/RUU/CS/techreps/CS-1997/

  2. W. Cook, T. Rutherford, H. E. Scarf, and D. Shallcross. An implementation of the generalized basis reduction algorithm for integer programming. ORSA Journal on Computing, 5:206–212, 1993.

    MATH  MathSciNet  Google Scholar 

  3. D. Coppersmith. Small solutions to polynomial equations, and low exponent RSA vulnerability. Journal of Cryptology, 10:233–260, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Cornuéjols, R. Urbaniak, R. Weismantel, and L. Wolsey. Decomposition of integer programs and of generating sets. In R. Burkard and G. Woeginger, editors, Algorithms — ESA’ 97, LNCS, Vol. 1284, pages 92–103. Springer-Verlag, 1997.

    Google Scholar 

  5. M. J. Coster, A. Joux, B. A. LaMacchia, A. M. Odlyzko, and C. P. Schnorr. Improved low-density subset sum algorithms. Computational Complexity, 2:111–128, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  6. CPLEX Optimization Inc. Using the CPLEX Callable Library, 1989.

    Google Scholar 

  7. B. de Fluiter. A Complexity Catalogue of High-Level Synthesis Problems. Master’s thesis, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1993.

    Google Scholar 

  8. G. Havas, B. S. Majewski, and K. R. Matthews. Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia, 1996.

    Google Scholar 

  9. J. C. Lagarias and A. M. Odlyzko. Solving low-density subset sum problems. Journal of the Association for Computing Machinery, 32:229–246, 1985.

    MATH  MathSciNet  Google Scholar 

  10. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538–548, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  12. LiDIA — A library for computational number theory. TH Darmstadt / Universität des Saarlandes, Fachbereich Informatik, Institut für Theoretische Informatik. http://www.informatik.th-darmstadt.de/pub/TI/LiDIA

  13. L. Lovász and H. E. Scarf. The generalized basis reduction algorithm. Mathematics of Operations Research, 17:751–764, 1992.

    MATH  MathSciNet  Google Scholar 

  14. M. C. McFarland, A. C. Parker, and R. Camposano. The high-level synthesis of digital systems. Proceedings of the IEEE, Vol. 78, pages 301–318, 1990.

    Article  Google Scholar 

  15. C. P. Schnorr and M. Euchner. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Mathematical Programming, 66:181–199, 1994.

    Article  MathSciNet  Google Scholar 

  16. C. P. Schnorr and H. H. Hörner. Attacking the Chor-Rivest Cryptosystem by improved lattice reduction. In L. C. Guillou and J.-J. Quisquater, editors, Advances in Cryptology — EUROCRYPT’ 95, LNCS, Vol. 921, pages 1–12. Springer Verlag, 1995.

    Google Scholar 

  17. W. F. J. Verhaegh, P. E. R. Lippens, E. H. L. Aarts, J. H. M. Korst, J. L. van Meerbergen, and A. van der Werf. Modeling periodicity by PHIDEO steams. Proceedings of the Sixth International Workshop on High-Level Synthesis, pages 256–266. ACM/SIGDA, IEEE/DATC, 1992.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Utrecht University, Utrecht

    Karen Aardal

  2. Department of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven

    Cor Hurkens

  3. Emerging Technology, Citibank, NA

    Arjen K. Lenstra

Authors
  1. Karen Aardal
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  2. Cor Hurkens
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  3. Arjen K. Lenstra
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Editor information

Editors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6020 Annapolis, Houston, TX, 77005, USA

    Robert E. Bixby

  2. PROS Strategic Solutions, 3223 Smith Street, Houston, TX, 77006, USA

    E. Andrew Boyd

  3. Department of Industrial, Engineering, Texas A&M University, 1000 Country Place Dr. Apt. 69, Houston, TX, 77079, USA

    Roger Z. Ríos-Mercado

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© 1998 Springer-Verlag Berlin Heidelberg

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Aardal, K., Hurkens, C., Lenstra, A.K. (1998). Solving a Linear Diophantine Equation with Lower and Upper Bounds on the Variables. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_18

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  • DOI: https://doi.org/10.1007/3-540-69346-7_18

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  • Print ISBN: 978-3-540-64590-0

  • Online ISBN: 978-3-540-69346-8

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