Abstract
In a space of dimension 30 we find a pair of parallel hyperplanes, uniquely determined by vertices of a unit cube lying on them, such that strictly between the hyperplanes there are no vertices of the cube, though there are integer points. A similar two-sided example is constructed in dimension 37. We consider possible locations of empty quadrics with respect to vertices of the cube, which is a particular case of a discrete optimization problem for a quadratic polynomial on the set of vertices of the cube. We demonstrate existence of a large number of pairs of parallel hyperplanes such that each pair contains a large number of points of a prescribed set.
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Beresnev, V.L., Diskretnye zadachi razmeshcheniya i polinomy ot bulevykh peremennykh (Discrete Distribution Problems and Polynomials in Boolean Variables), Novosibirsk: Inst. Mat., 2005.
Billionnet, A. and Elloumi, S., Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0–1 Problem, Math. Program., Ser. A, 2007, vol. 109, no. 1, pp. 55–68.
Ahlatçioğlu, A., Bussieck, M., Esen, M., Guignard, M., Jagla, J.-H., and Meeraus, A., Combining QCR and CHR for Convex Quadratic Pure 0-1 Programming Problems with Linear Constraints, Ann. Oper. Res. (online publ.), September 30, 2011, DOI:10.1007/s10479-011-0969-1.
Emelichev, V.A. and Korotkov, V.V., On Stability Radius of Effective Solution of Vector Quadratic Boolean Bottleneck Problem, Diskretn. Anal. Issled. Oper., 2011, vol. 18, no. 6, pp. 3–16.
Seliverstov, A.V. and Lyubetsky, V.A., On Symmetric Matrices with Indeterminate Leading Diagonals, Probl. Peredachi Inf., 2009, vol. 45, no. 3, pp. 73–78 [Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 3, pp. 258–263].
Erdős, P., On a Lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 1945, vol. 51, no. 12, pp. 898–902.
Costello, K.P. and Vu, V.H., The Rank of Random Graphs, Random Structures and Algorithms, 2008, vol. 33, no. 3, pp. 269–285.
Shnurnikov, I.N., Cardinality of a Separable Set of Vertices of a Multidimensional Cube, Vestnik Moskov. Univ., Ser. I: Mat. Mekh., 2010, no. 2, pp. 11–17.
Seliverstov, A.V. and Lyubetsky, V.A., On Forms Vanishing at Every Vertex of a Cube, Inform. Protsessy, 2011, vol. 11, no. 3, pp. 330–335.
Beresnev, V.L., Goncharov, E.N., and Mel’nikov, A.A., Local Search over a Generalized Neighborhood for a Problem of Optimizing Pseudo-Boolean Functions, Diskretn. Anal. Issled. Oper., 2011, vol. 18, no. 4, pp. 3–16.
Alon, N. and Füredi, Z., Covering the Cube by Affine Hyperplanes, European J. Combin., 1993, vol. 14, no. 2, pp. 79–83.
Ibarra, O.H. and Kim, C.E., Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems, J. Assoc. Comput. Mach., 1975, vol. 22, no. 4, pp. 463–468.
Schrijver, A., Theory of Linear and Integer Programming, Chichester: Wiley, 1986. Translated under the title Teoriya lineinogo i tselochislennogo programmirovaniya, Moscow: Mir, 1991.
Xiao, Q.-F., Hu, X.-Y., and Zhang, L., The Symmetric Minimal Rank Solution of the Matrix Equation AX = B and the Optimal Approximation, Electron. J. Linear Algebra, 2009, vol. 18, pp. 264–273.
Mitra, S.K., Fixed Rank Solutions of Linear Matrix Equations, Sankhyā, Ser. A, 1972, vol. 34, no. 4, pp. 387–392.
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Original Russian Text © K.Yu. Gorbunov, A.V. Seliverstov, V.A. Lyubetsky, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 2, pp. 113–120.
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Gorbunov, K.Y., Seliverstov, A.V. & Lyubetsky, V.A. Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube. Probl Inf Transm 48, 185–192 (2012). https://doi.org/10.1134/S0032946012020081
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DOI: https://doi.org/10.1134/S0032946012020081