FPT-algorithms for some problems related to integer programming
D.V. Gribanov, D.S. Malyshev, P.M. Pardalos, S.I. Veselov
TL;DR
This paper develops fixed-parameter tractable algorithms for specific integer programming problems with near square matrices, focusing on cases where matrices have bounded minors or no singular sub-matrices, advancing computational methods in this domain.
Contribution
It introduces new FPT-algorithms for integer programming problems based on matrix properties like rank minors and singular sub-matrices, expanding algorithmic options for these cases.
Findings
FPT-algorithms for shortest lattice vector with near square matrices
FPT-algorithms for integer linear programming with bounded minors
FPT-algorithms for simplex width computation in special matrix cases
Abstract
In this paper, we present FPT-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 rank minors of the corresponding matrices. Additionally, we present FPT-algorithms with respect to the same parameter for the problems, when the matrices have no singular rank sub-matrices.
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