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 \(n\ge 2\).
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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)
Antony, M., Brightwell, G., Shawe-Taylor, J.: On exact specification by labelled examples. Discrete Appl. Math. 61(1), 1–25 (1995)
Angluin, D.: Queries and concept learning. Discrete Appl. Math. 2, 319–342 (1988)
Bárány, I., Howe, R., Lovász, L.: On integer points in polyhedra: a lower bound. Combinatorica 12(2), 135–142 (1992)
Chirkov, A.Yu.: Caratheodory’s theorem and coverings of a polyhedron by simplexes. Manuscript No. 668B93, deposited at VINITI, Moscow, (1993). (in Russian)
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)
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)
Cook, W., Hartmann, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12(1), 27–37 (1992)
Håstad, J.: On the size of weights for threshold gates. SIAM J. Discrete Math. 7(3), 484–492 (1994)
Hayes, A.S., Larman, D.C.: The vertices of the knapsack polytope. Discrete Appl. Math. 6(2), 135–138 (1983)
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)
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)
McMullen, P.: The maximum number of faces of a convex polytope. Mathematika 17, 179–184 (1991)
Morgan, D.A.: Upper and lower bound results on the convex hull of integer points in polyhedra. Mathematika 38(2), 321–328 (1991)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience, New York (1986)
Shevchenko, V.N.: On the number of extreme points in integer programming. Kibernetika 2, 133–134 (1981)
Shevchenko, V.N., Zolotykh, N.Yu.: On complexity of deciphering threshold functions of \(k\)-valued logic. Russ. Math. Dokl. 362(5), 606–608 (1998)
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)
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)
Ziegler, G.M.: Lectures on Polytopes. Springer, Berlin (1995)
Zolotykh, N.Yu.: On the number of vertices in integer linear programming problems. arXiv:math/0611356 [math.CO] (2006)
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)
Zolotykh, N.Yu., Shevchenko, V.N.: On complexity of deciphering threshold functions. Discrete Anal. Oper. Res. 2(3), 72–73 (1995). (in Russian)
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)
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.
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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
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DOI: https://doi.org/10.1007/s00373-016-1683-1