Skip to main content
SpringerLink
Log in

Proof of covering minimality by generalizing the notion of independence

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

A method is proposed for obtaining lower bounds for the length of the shortest cover and complexity of the minimal cover based on the notion of independent family of sets. For the problem of minimization of Boolean functions, we provide the functions and construct coverings by faces of the set of unit vertices for which the suggested lower bounds can be achieved in the case of five or more variables. The lower bounds, based on independent sets, are unreachable and cannot be used as sufficient conditions for minimality of such functions.

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. A. V. Eremeev, L. A. Zaozerskaya, and A. A. Kolokolov, “The Set Covering Problem: Complexity, Algorithms, and Experimental Investigations,” Diskretn. Anal. Issled. Oper. Ser. 2, 7 (2), 22–46 (2000).

    MathSciNet  MATH  Google Scholar 

  2. G. I. Zabinyako, “Implementation of Algorithms for Solution of Covering Problems and Analysis of Their Efficiency,” Vychisl. Technol. 12 (6), 50–58 (2007) [Comput. Technol. 12 (6), 50–58 (2007)].

    MATH  Google Scholar 

  3. V. K. Leont’ev, “Discrete optimization,” Zh. Vychisl. Mat. Mat. Fiz. 47 (2), 338–352 (2007) [Comput. Math. Math. Phys. 47 (2), 328–340 (2007)].

    MathSciNet  MATH  Google Scholar 

  4. I. P. Chukhrov, “On Complexity Measures of Complexes of Faces in the Unit Cube, Diskretn. Anal. Issled. Oper. 20 (6), 77–94 (2013) [J. Appl. Indust. Math. 8 (1), 9–19 (2014)].

  5. I. P. Chukhrov, “On a Minimization Problem for a Set of Boolean Functions,” Diskretn. Anal. Issled. Oper. 22 (3), 75–97 (2015) [J. Appl. Indust. Math. 9 (3), 335–350 (2015)].

    MathSciNet  MATH  Google Scholar 

  6. S. Al-Shihabi, M. Arafeh, and M. Barghash, “An Improved Hybrid Algorithm for the Set Covering Problem,” Comput. Ind. Eng. 85, 328–334 (2015).

    Article  Google Scholar 

  7. O. Coudert, “On SolvingCovering Problems,” in Proceedings of the 33rdDesign Automation Conference, Las Vegas, NV, USA, June 3–7, 1996 (ACM, New York, 1996), pp. 197–202.

    Google Scholar 

  8. O. Coudert and T. Sasao, “Two-Level Logic Minimization,” in Logic Synthesis and Verification, Ed. by S. Hassoun and T. Sasao (Kluwer Acad. Publ., Norwell, MA, 2002), pp. 1–27.

    Chapter  Google Scholar 

  9. C. Gao, X. Yao, T. Weise, and J. Li, “An Efficient Local Search Heuristic with RowWeighting for the Unicost Set Covering Problem,” Europ. J. Oper. Res. 246 (3), 750–761 (2015).

    Article  MATH  Google Scholar 

  10. N. Sapkota and C. H. Reilly, “Simulating Realistic Set Covering Problems with Known Optimal Solutions,” Comput. Ind. Eng. 61 (1), 39–47 (2011).

    Article  Google Scholar 

  11. F. J. Vasko, Y. Lu, and K. Zyma, “What Is the Best Greedy-Like Heuristic for the Weighted Set Covering Problem?”Oper. Res. Lett. 44 (3), 366–369 (2016).

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Aided Design, Vtoraya Brestskaya ul. 19/18, Moscow, 123056, Russia

    I. P. Chukhrov

Authors
  1. I. P. Chukhrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. P. Chukhrov.

Additional information

Original Russian Text © I.P. Chukhrov, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 2, pp. 87–106.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chukhrov, I.P. Proof of covering minimality by generalizing the notion of independence. J. Appl. Ind. Math. 11, 193–203 (2017). https://doi.org/10.1134/S1990478917020053

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478917020053

Keywords

Search

Navigation