Abstract
We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to a linear program max{wx: Ax≤b}, the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixA. Using this, we strengthen several integer programming ‘proximity’ results of Blair and Jeroslow; Graver; and Wolsey. We also show that the Chvátal rank of a polyhedron {x: Ax≤b} can be bounded above by a function of the matrixA, independent of the vectorb, a result which, as Blair observed, is equivalent to Blair and Jeroslow's theorem that ‘each integer programming value function is a Gomory function.’
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Supported by a grant from the Alexander von Humboldt Stiftung.
Since September 1985: Department of Operations Research, Upson Hall, Cornell University, Ithaca, NY 14853, USA.
Partially supported by the Sonderforschungbereich 21 (DFG), Institut für Ökonometrie und Operations Research of the University of Bonn, FR Germany.
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Cook, W., Gerards, A.M.H., Schrijver, A. et al. Sensitivity theorems in integer linear programming. Mathematical Programming 34, 251–264 (1986). https://doi.org/10.1007/BF01582230
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DOI: https://doi.org/10.1007/BF01582230