Skip to main content
SpringerLink
Log in

An improved algorithm to compute the \(\omega \)-primality

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we present an improved methodology to compute the \(\omega \)-primality of a numerical semigroup. The approach is based on exploiting the structure of the problem on a resolution method for optimizing a linear function over the set of efficient solutions of a multiple objective integer linear programming problem. The numerical experiments show the efficiency of the proposed technique compared to the existing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ehrgott, M.: Multicriteria Optimization, vol. 491. Springer, Berlin (2013)

    MATH  Google Scholar 

  2. Gandibleux, X.: Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys, vol. 52. Springer, Berlin (2002)

    MATH  Google Scholar 

  3. Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Applications. Wiley, New York (1986)

    MATH  Google Scholar 

  4. Shigeno, M., Takahashi, I., Yamamoto, Y.: Minimum maximal flow problem: an optimization over the efficient set. J. Glob. Optim. 25(4), 425–443 (2003)

    Article  MathSciNet  Google Scholar 

  5. Benson, H.P.: Optimization over the efficient set. J. Math. Anal. Appl. 98(2), 562–580 (1984)

    Article  MathSciNet  Google Scholar 

  6. Ecker, J.G., Song, J.H.: Optimizing a linear function over an efficient set. J. Optim. Theory Appl. 83(3), 541–563 (1994)

    Article  MathSciNet  Google Scholar 

  7. Philip, J.: Algorithms for the vector maximization problem. Math. Program. 2(1), 207–229 (1972)

    Article  MathSciNet  Google Scholar 

  8. Sayin, S.: Optimizing over the efficient set using a top-down search of faces. Oper. Res. 48(1), 65–72 (2000)

    Article  MathSciNet  Google Scholar 

  9. Yamamoto, Y.: Optimization over the efficient set: overview. J. Glob. Optim. 22(1–4), 285–317 (2002)

    Article  MathSciNet  Google Scholar 

  10. Djamal, C., Moncef, A.: Optimizing a linear function over an integer efficient set. Eur. J. Oper. Res. 174(2), 1140–1161 (2006)

    Article  MathSciNet  Google Scholar 

  11. Boland, N., Charkhgard, H., Savelsbergh, M.: A new method for optimizing a linear function over the efficient set of a multiobjective integer program. Eur. J. Oper. Res. 260, 904–919 (2016)

    Article  MathSciNet  Google Scholar 

  12. Djamal, M.F.C.: Optimization of a linear function over the set of stochastic efficient solutions. Comput. Manag. Sci. 11, 157–178 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jorge, J.M.: An algorithm for optimizing a linear function over an integer efficient set. Eur. J. Oper. Res. 195(1), 98–103 (2009)

    Article  MathSciNet  Google Scholar 

  14. Geroldinger, A., Halter-Koch, F.: Non-unique Factorizations: A Survey. Springer, Berlin (2006)

    Book  Google Scholar 

  15. Barron, T., O’Neill, C., Pelayo, R.: On dynamic algorithms for factorization invariants in numerical monoids. Math. Comput. 86(307), 2429–2447 (2017)

    Article  MathSciNet  Google Scholar 

  16. Blanco, V., García-Sánchez, P.A., Geroldinger, A.: Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and krull monoids. Ill. J. Math. 55(4), 1385–1414 (2011)

    Article  MathSciNet  Google Scholar 

  17. García-García, J.I., Moreno-Frías, M.A., Vigneron-Tenorio, A.: Computation of the \(\omega \)-primality and asymptotic \(\omega \)-primality with applications to numerical semigroups. Isr. J. Math. 206(1), 395–411 (2015)

    Article  MathSciNet  Google Scholar 

  18. García-Sánchez, P.A., O’Neill, C., Webb, G.: The computation of factorization invariants for affine semigroups. J. Algebra Appl. 18(01), 1950019 (2019)

    Article  MathSciNet  Google Scholar 

  19. Halter-Koch, F.: The tame degree and related invariants of non-unique factorizations. Acta Math. Univ. Ostrav. 16(1), 57–68 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Omidali, M.: The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math. 24, 627–640 (2012)

    Article  MathSciNet  Google Scholar 

  21. O’Neill, C., Pelayo, R.: How do you measure primality? Am. Math. Mon. 122(2), 121–137 (2015)

    Article  MathSciNet  Google Scholar 

  22. O’Neill, C., Pelayo, R.: Factorization invariants in numerical monoids? Algebraic Geom. Methods Discrete Math. 685, 231–250 (2017)

    Article  MathSciNet  Google Scholar 

  23. Blanco, V.: A mathematical programming approach to the computation of the omega invariant of a numerical semigroup. Eur. J. Oper. Res. 215(3), 539–550 (2011)

    Article  MathSciNet  Google Scholar 

  24. Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization. Kluwer, Boston (2002)

    MATH  Google Scholar 

  25. Ecker, J.G., Kouada, I.A.: Finding efficient points for linear multiple objective programs. Math. Program. 8(1), 375–377 (1975)

    Article  MathSciNet  Google Scholar 

  26. Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups, vol. 20. Springer, Berlin (2009)

    Book  Google Scholar 

  27. Geroldinger, A., Hassler, W.: Arithmetic of mori domains and monoids. J. Algebra 319(8), 3419–3463 (2008)

    Article  MathSciNet  Google Scholar 

  28. Geroldinger, A., Hassler, W.: Local tameness of v-noetherian monoids. J. Pure Appl. Algebra 212(6), 1509–1524 (2008)

    Article  MathSciNet  Google Scholar 

  29. Anderson, D.F., Scott T, C.: How far is an element from being prime? J. Algebra Appl. 9(05), 779–789 (2010)

    Article  MathSciNet  Google Scholar 

  30. Anderson, D.F., Chapman, S.T., Kaplan, N., Torkornoo, D.: An algorithm to compute \(\omega \)-primality in a numerical monoid. Semigroup Forum 82, 96–108 (2011)

    Article  MathSciNet  Google Scholar 

  31. Geroldinger, A., Hassler, W., Lettl, G.: On the arithmetic of strongly primary monoids. Semigroup Forum 75, 567–587 (2007)

    Article  MathSciNet  Google Scholar 

  32. Blanco, V., Rosales, J.C.: Irreducibility in the set of numerical semigroups with fixed multiplicity. Int. J. Algebra Comput. 21(5), 731–744 (2011)

    Article  MathSciNet  Google Scholar 

  33. Blanco, V., García-Sánchez, P.A., Puerto, J.: Counting numerical semigroups with short generating functions. Int. J. Algebra Comput. 21(7), 1217–1235 (2011)

    Article  MathSciNet  Google Scholar 

  34. Blanco, V., Puerto, J.: An application of integer programming to the decomposition of numerical semigroups. SIAM J. Discrete Math. 26(3), 1210–1237 (2012)

    Article  MathSciNet  Google Scholar 

  35. Blanco, V., Rosales, J.C.: The set of numerical semigroups of a given genus. Semigroup Forum 85(2), 255–267 (2012)

    Article  MathSciNet  Google Scholar 

  36. Blanco, V., Rosales, J.C.: On the enumeration of the set of numerical semigroups with fixed Frobenius number. Comput. Math. Appl. 63(7), 1204–1211 (2012)

    Article  MathSciNet  Google Scholar 

  37. Blanco, V., Rosales, J.C.: The tree of irreducible numerical semigroups with fixed Frobenius number. Forum Math. 25, 1249–1261 (2013)

    Article  MathSciNet  Google Scholar 

  38. Sylva, J., Crema, A.: A method for finding the set of non-dominated vectors for multiple objective integer linear programs. Eur. J. Oper. Res. 158(1), 46–55 (2004)

    Article  MathSciNet  Google Scholar 

  39. Sylva, J., Crema, A.: A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs. Eur. J. Oper. Res. 180(3), 1011–1027 (2007)

    Article  MathSciNet  Google Scholar 

  40. Delgado, M., García-Sánchez, P.A.: numericalsgps, a GAP package for numerical semigroups. ACM Commun. Comput. Algebra 50, 12–24 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The third author was partially supported by the Projects MTM2016-74983-C2-1-R (MINECO, Spain), PP2016-PIP06 (Universidad de Granada) and the research group SEJ-534 (Junta de Andalucía).

Author information

Authors and Affiliations

  1. Department of Operations Research, USTHB, Algiers, Algeria

    Wissem Achour & Djamal Chaabane

  2. IEMath-GR, Universidad de Granada, Granada, Spain

    Víctor Blanco

Authors
  1. Wissem Achour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Djamal Chaabane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Víctor Blanco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Víctor Blanco.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Achour, W., Chaabane, D. & Blanco, V. An improved algorithm to compute the \(\omega \)-primality. Optim Lett 15, 97–107 (2021). https://doi.org/10.1007/s11590-020-01589-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01589-w

Keywords

Search

Navigation