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On integer programming with bounded determinants

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Abstract

Let A be an \((m \times n)\) integral matrix, and let \(P=\{ x :A x \le b\}\) be an n-dimensional polytope. The width of P is defined as \( w(P)=min\{ x\in \mathbb {Z}^n{\setminus }\{0\} :max_{x \in P} x^\top u - min_{x \in P} x^\top v \}\). Let \(\varDelta (A)\) and \(\delta (A)\) denote the greatest and the smallest absolute values of a determinant among all \(r(A) \times r(A)\) sub-matrices of A, where r(A) is the rank of the matrix A. We prove that if every \(r(A) \times r(A)\) sub-matrix of A has a determinant equal to \(\pm \varDelta (A)\) or 0 and \(w(P)\ge (\varDelta (A)-1)(n+1)\), then P contains n affine independent integer points. Additionally, we present similar results for the case of k-modular matrices. The matrix A is called totally k-modular if every square sub-matrix of A has a determinant in the set \(\{0,\, \pm k^r :r \in \mathbb {N} \}\). When P is a simplex and \(w(P)\ge \delta (A)-1\), we describe a polynomial time algorithm for finding an integer point in P.

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Acknowledgments

The author wishes to express special thanks for the invaluable assistance to P. M. Pardalos, A. Y. Chirkov, D. S. Malyshev, N. Y. Zolotykh and V. N. Shevchenko. The work is supported by LATNA Laboratory NRU HSE RF government Grant Ag. 11.G34.31.0057 and Russian Foundation for Basic Research Grant Ag. 15-01-06249 A.

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603950, Russian Federation

    D. V. Gribanov & S. I. Veselov

  2. Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, 136 Rodionova, Nizhny Novgorod, 603093, Russian Federation

    D. V. Gribanov

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  1. D. V. Gribanov
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  2. S. I. Veselov
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Correspondence to D. V. Gribanov.

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Gribanov, D.V., Veselov, S.I. On integer programming with bounded determinants. Optim Lett 10, 1169–1177 (2016). https://doi.org/10.1007/s11590-015-0943-y

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