Advertisement

A subdivision algorithm to reason on high-degree polynomial constraints over finite domains

  • Federico BergentiEmail author
  • Stefania Monica
Article
  • 2 Downloads

Abstract

This paper proposes an algorithm to reason on constraints expressed in terms of polynomials with integer coefficients whose variables take values from finite subsets of the integers. The proposed algorithm assumes that an initial approximation of the domains of variables is available in terms of a bounding box, and it recursively subdivides the box into disjoint boxes until a termination condition is met. The algorithm includes three termination conditions that allow using it for three related reasoning tasks: constraint satisfaction, enumeration of solutions, and hyper-arc consistency enforcement. Considered termination conditions are based on suitable lower and upper bounds for polynomial functions over boxes that are determined using new results proved in the paper. The algorithm is particularly appropriate to reason on high-degree polynomial constraints because the proposed method to determine lower and upper bounds can outperform alternative methods when high-degree polynomials in a moderate number of variables are considered.

Keywords

High-degree polynomial constraints Polynomial constraints over finite domains Constraint satisfaction problems 

Mathematics Subject Classification

68T01 68T27 68T99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Apt, K.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  2. 2.
    Bergenti, F., Monica, S.: Hyper-arc consistency of polynomial constraints over finite domains using the modified Bernstein form. Ann. Math. Artif. Intell. 80(2), 131–151 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergenti, F., Monica, S.: Satisfaction of polynomial constraints over finite domains using function values. In: Della Monica, D., Murano, A., Rubin, S., Sauro, L. (eds.) Joint Proceedings of the 18th Italian Conference on Theoretical Computer Science and the 32nd Italian Conference on Computational Logic (ICTCS 2017 and CILC 2017), CEUR Workshop Proceedings, vol. 1949, pp 262–275. RWTH Aachen (2017)Google Scholar
  4. 4.
    Bergenti, F., Monica, S.: Simple and effective sign consistency using interval arithmetic. In: Casagrande, A., Omodeo, E.G. (eds.) Proceedings of the 34th Italian Conference on Computational Logic (CILC 2019), CEUR Workshop Proceedings, vol. 2396, pp 89–103. RWTH Aachen (2019)Google Scholar
  5. 5.
    Bergenti, F., Monica, S., Rossi, G.: Polynomial constraint solving over finite domains with the modified Bernstein form. In: Fiorentini, C., Momigliano, A. (eds.) Proceedings of the 31st Italian Conference on Computational Logic (CILC 2016), CEUR Workshop Proceedings, vol. 1645, pp 118–131. RWTH Aachen (2016)Google Scholar
  6. 6.
    Bergenti, F., Monica, S., Rossi, G.: A subdivision approach to the solution of polynomial constraints over finite domains using the modified Bernstein form. In: Adorni, G., Cagnoni, S., Gori, M., Maratea, M. (eds.) AI*IA 2016 Advances in Artificial Intelligence, Lecture Notes in Computer Science, vol. 10037, pp 179–191. Springer International Publishing (2016)Google Scholar
  7. 7.
    Bergenti, F., Monica, S., Rossi, G.: Constraint logic programming with polynomial constraints over finite domains. Fundamenta Informaticae 161(1–2), 9–27 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bhansali, S., Kramer, G.A., Hoar, T.J.: A principled approach towards symbolic geometric constraint satisfaction. J. Artif. Intell. Res. 4, 419–443 (1996)CrossRefGoogle Scholar
  9. 9.
    Farouki, R.T.: The Bernstein polynomial basis: A centennial retrospective. Comput.-Aided Geom. Des. 29(6), 379–419 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Comput.-Aided Geom. Des. 5(1), 1–26 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garloff, J.: Convergent bounds for the range of multivariate polynomials. In: Nickel, K. (ed.) Interval Mathematics 1985, Lecture Notes in Computer Science, vol. 212, pp 37–56. Springer International Publishing (1986)Google Scholar
  12. 12.
    Garloff, J., Smith, A.P.: Solution of systems of polynomial equations by using Bernstein expansion. In: Alefeld, G., Rohn, J., Rump, S., Yamamoto, T. (eds.) Symbolic Algebraic Methods and Verification Methods, pp 87–97. Springer International Publishing (2001)Google Scholar
  13. 13.
    Grimstad, B., Sandnes, A.: Global optimization with spline constraints: A new branch-and-bound method based on B-splines. J. Glob. Optim. 65(3), 401–439 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lorentz, G.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953)zbMATHGoogle Scholar
  15. 15.
    Malapert, A., Régin, J.C., Rezgui, M.: Embarrassingly parallel search in constraint programming. J. Artif. Intell. Res. 57, 421–464 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mourrain, B., Pavone, J.: Subdivision methods for solving polynomial equations. J. Symb. Comput. 44(3), 292–306 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nataraj, P., Arounassalame, M.: A new subdivision algorithm for the Bernstein polynomial approach to global optimization. Int. J. Autom. Comput. 4(4), 342–352 (2007)CrossRefGoogle Scholar
  18. 18.
    Peña, J. M., Sauer, T.: On the multivariate Horner scheme. SIAM J. Numer. Anal. 37(4), 1186–1197 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ray, S., Nataraj, P.: An efficient algorithm for range computation of polynomials using the Bernstein form. J. Glob. Optim. 45, 403–426 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ray, S., Nataraj, P.: A matrix method for efficient computation of Bernstein coefficients. Reliab. Comput. 17, 40–71 (2012)MathSciNetGoogle Scholar
  21. 21.
    Rivlin, T.J.: Bounds on a polynomial. J. Res. Natl. Bur. Stand. 74B(1), 47–54 (1970)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rossi, F., van Beek, P., Walsh, T.: Handbook of Constraint Programming. Elsevier, New York (2006)zbMATHGoogle Scholar
  23. 23.
    Sánchez-Reyes, J.: Algebraic manipulation in the Bernstein form made simple via convolutions. Comput. Aided Des. 35, 959–967 (2003)CrossRefGoogle Scholar
  24. 24.
    Smith, A.P.: Fast construction of constant bound functions for sparse polynomials. J. Glob. Optim. 43(2), 445–458 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Steffens, K.G.: The History of Approximation Theory: From Euler to Bernstein. Birkhäuser, Boston (2006)zbMATHGoogle Scholar
  26. 26.
    Titi, J., Garloff, J.: Matrix methods for the tensorial Bernstein form. Appl. Math. Comput. 346, 254–271 (2019)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Triska, M.: The finite domain constraint solver of SWI-Prolog. In: Schrijvers, T., Thiemann, P. (eds.) Functional and Logic Programming, Lecture Notes in Computer Science, vol. 7294, pp 307–316. Springer International Publishing (2012)Google Scholar
  28. 28.
    Wielemaker, J., Schrijvers, T., Triska, M., Lager, T.: SWI-Prolog. Theory Practice Logic Program. 12(1–2), 67–96 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche, Fisiche e InformaticheUniversità degli Studi di ParmaParmaItaly

Personalised recommendations