Skip to main content
SpringerLink
Log in

On the Number of Irreducible Points in Polyhedra

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in n-dimensional polytope P is at most \(O(m^{\lfloor \frac{n}{2}\rfloor }\log ^{n-1}\gamma )\), where n is fixed and P is given by a system of m linear inequalities with integer coefficients not exceeding (by absolute value) \(\gamma \). This bound is tight. Using this result we prove the conjecture asserting that the teaching dimension in the class of threshold functions of k-valued logic in n variables is \(\varTheta (\log ^{n-2} k)\) for any fixed \(n\ge 2\).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Alekseyev, M.A., Basova, M., Zolotykh, N.Yu.: On the minimal teaching sets of two-dimensional threshold functions. SIAM J. Discrete Math. 29(1), 157–165 (2015)

  2. Antony, M., Brightwell, G., Shawe-Taylor, J.: On exact specification by labelled examples. Discrete Appl. Math. 61(1), 1–25 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Angluin, D.: Queries and concept learning. Discrete Appl. Math. 2, 319–342 (1988)

    MathSciNet  Google Scholar 

  4. Bárány, I., Howe, R., Lovász, L.: On integer points in polyhedra: a lower bound. Combinatorica 12(2), 135–142 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chirkov, A.Yu.: Caratheodory’s theorem and coverings of a polyhedron by simplexes. Manuscript No. 668B93, deposited at VINITI, Moscow, (1993). (in Russian)

  6. Chirkov, A.Yu.: On the lower bound for the number of vertices of convex hull of integer and partially integer points of a polyhedron. Discrete Anal. Oper. Res. 3(2), 80–89 (1996)

  7. Chirkov, A.Yu.: The relationship between upper bounds of the number of vertices of convex hull of integer points of a polyhedron and its metric characteristics. In: Proceedings of the First International Conference on Mathematical Algorithms, pp. 169–174. Nizhni Novgorod State University Publisher (1997). (in Russian)

  8. Cook, W., Hartmann, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12(1), 27–37 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Håstad, J.: On the size of weights for threshold gates. SIAM J. Discrete Math. 7(3), 484–492 (1994)

    Article  MathSciNet  Google Scholar 

  10. Hayes, A.S., Larman, D.C.: The vertices of the knapsack polytope. Discrete Appl. Math. 6(2), 135–138 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hegedüs, T.: Geometrical concept learning and convex polytopes. In: Proceedings of the 7th Annual ACM Conference on Computational Learning Theory (COLT’94) pp. 228–236. New York, ACM Press (1994)

  12. Hegedüs, T.: Generalized teaching dimensions and the query complexity of learning. In: Proceedings of the 8th Annual ACM Conference on Computational Learning Theory (COLT’95) pp. 108–117. New York, ACM Press (1995)

  13. McMullen, P.: The maximum number of faces of a convex polytope. Mathematika 17, 179–184 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Morgan, D.A.: Upper and lower bound results on the convex hull of integer points in polyhedra. Mathematika 38(2), 321–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience, New York (1986)

    MATH  Google Scholar 

  16. Shevchenko, V.N.: On the number of extreme points in integer programming. Kibernetika 2, 133–134 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Shevchenko, V.N., Zolotykh, N.Yu.: On complexity of deciphering threshold functions of \(k\)-valued logic. Russ. Math. Dokl. 362(5), 606–608 (1998)

  18. Veselov, S.I.: A lower bound for the mean number of irreducible and extreme points in two discrete programming problems. Manuscript 61984, deposited at VINITI, Moscow (1984). (in Russian)

  19. Veselov, S.I., Chirkov, A.Yu.: Some estimates for the number of vertices of integer polyhedra. J. Appl. Ind. Math. 2(4), 591–604 (2008)

  20. Ziegler, G.M.: Lectures on Polytopes. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  21. Zolotykh, N.Yu.: On the number of vertices in integer linear programming problems. arXiv:math/0611356 [math.CO] (2006)

  22. Zolotykh, N.Yu., Chirkov, A.Y.: On the upper bound for cardinality of the minimal teaching set of a threshold function. Discrete Anal. Oper. Res. 19(5), 35–46 (2012). (in Russian)

  23. Zolotykh, N.Yu., Shevchenko, V.N.: On complexity of deciphering threshold functions. Discrete Anal. Oper. Res. 2(3), 72–73 (1995). (in Russian)

  24. Zolotykh, N.Yu., Shevchenko, V.N.: Estimating the complexity of deciphering a threshold functions in a \(k\)-valued logic. Comput. Math. Math. Phys. 39(2), 328–334 (1999)

Download references

Acknowledgments

The authors thank V. N. Shevchenko and S. I. Veselov for fruitfull discussions and referees for very useful suggestions. The work is partially supported by Russian Foundation for Basic Research, Grant Number 15-01-06249.

Author information

Authors and Affiliations

  1. National Research Lobachevsky State University of Nizhni Novgorod, 23 Prospekt Gagarina, Nizhny Novogorod, Russia, 603950

    Aleksandr Yu. Chirkov & Nikolai Yu. Zolotykh

  2. National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhni Novgorod, Russia, 603155

    Nikolai Yu. Zolotykh

Authors
  1. Aleksandr Yu. Chirkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolai Yu. Zolotykh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolai Yu. Zolotykh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chirkov, A.Y., Zolotykh, N.Y. On the Number of Irreducible Points in Polyhedra. Graphs and Combinatorics 32, 1789–1803 (2016). https://doi.org/10.1007/s00373-016-1683-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-016-1683-1

Keywords

Search

Navigation