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The Flatness Theorem for Some Class of Polytopes and Searching an Integer Point

  • Dmitry V. GribanovEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 104)

Abstract

Let A be an m × n integral matrix of the rank n, we say that A has bounded minors if the maximum of the absolute values of the n × n minors is at most k, we will call these matrices as k-modular. We investigate an integer program \(\mathit{max}\{c^{\top }x: Ax \leq b,x \in \mathbb{Z}^{n}\}\), where A is k-modular. We say that A is almost unimodular if it is 2-modular and the absolute values of its \((n - 1) \times (n - 1)\) minors are at most 1. We also refer 2-modular matrices to as bimodular. We say that A is strict k-modular if the absolute values of its n × n minors are from set {0, k, −k}. We prove that the width of an empty lattice polytope is less than \((k - 1)\,(n + 1)\) if it is induced by a system of inequalities with a strict k-modular matrix. Furthermore, we can give a polynomial-time algorithm for searching an integer point in a strict k-modular polytope if its width is grater than \((k - 1)\,(n + 1)\).

Keywords

Convex Hull Interior Point Integer Program Convex Body Normal Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author wishes to express special thanks for the invaluable assistance to A.J. Chirkov, S.I. Veselov, D.S. Malyshev, V.N. Shevchenko, and S.V. Sorochan. The work is partly supported by National Research University Higher School of Economics, Russian Federation Government grant, N. 11.G34.31.0057.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Nizhny Novgorod Lobachevsky State UniversityNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation

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