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FPT-algorithms for some problems related to integer programming

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Abstract

In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems’ formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms with respect to the same parameter for the problems, when the matrices have no singular rank submatrices.

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Acknowledgements

Results of Sect. 3 were obtained under financial support of Russian Science Foundation Grant No. 14-41-00039. Results of Sect. 4 were obtained under financial support of Russian Science Foundation Grant No. 17-11-01336. Results of Sect. 5 were obtained under financial support of Russian Foundation for Basic Research, Grant No. 16-31-60008-mol-a-dk, and LATNA laboratory, NRU HSE.

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Authors and Affiliations

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

    D. V. Gribanov & S. I. Veselov

  2. National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod, Russian Federation, 603155

    D. V. Gribanov, D. S. Malyshev & P. M. Pardalos

  3. University of Florida, 401 Weil Hall, P.O. Box 116595, Gainesville, FL, 326116595, USA

    P. M. Pardalos

Authors
  1. D. V. Gribanov
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  2. D. S. Malyshev
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  3. P. M. Pardalos
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  4. S. I. Veselov
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Correspondence to D. S. Malyshev.

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Gribanov, D.V., Malyshev, D.S., Pardalos, P.M. et al. FPT-algorithms for some problems related to integer programming. J Comb Optim 35, 1128–1146 (2018). https://doi.org/10.1007/s10878-018-0264-z

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