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- Dmitry V. Gribanov, Sergey I. Veselov
- Optimization Letters
- 2016

- Dmitry V. Gribanov, Aleksandr Yu. Chirkov
- Optimization Letters
- 2016

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their… (More)

- Dmitry V. Gribanov, Dmitriy S. Malyshev
- Discrete Applied Mathematics
- 2017

- Dmitry V. Gribanov, Sergey I. Veselov
- ArXiv
- 2015

Let A be an (m × n) integral matrix, and let P = {x : Ax ≤ b} be an n-dimensional polytope. The width of P is defined as w(P ) = min{x ∈ Z\{0} : maxx∈Pxu−minx∈Pxv}. Let ∆(A) and δ(A) denote the greatest and the smallest absolute values of a determinant among all r(A)× r(A) submatrices of A, where r(A) is the rank of a matrix A. We prove that if every r(A) ×… (More)

Let A be an m × n integral matrix of rank n. We say that A has bounded minors if the maximum of the absolute values of the n × n minors is at most k, where k is a some natural constant. We will call that matrices like k-modular. We investigate an integer program max{cx : Ax ≤ b, x ∈ Zn} where A is k-modular. We say that A is almost unimodular (see [2, 5])… (More)

- D. V. Gribanov
- 2017

In this paper, we present FPT-algorithms for special cases of the shortest vector problem (SVP) and the integer linear programming problem (ILP), when matrices included to the problems’ formulations are near square. The main parameter is the maximal absolute value of rank minors of matrices included to the problem formulation. Additionally, we present… (More)

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