On sub-determinants and the diameter of polyhedra

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we show that the diameter of P is bounded by O(\Delta^2 n^4 log n\Delta). If P is bounded, then we show that the diameter of P is at most O(\Delta^2 n^3.5 log n\Delta). For the special case in which A is a totally unimodular matrix, the bounds are O(n^4 log n) and O(n^3.5 log n) respectively. This improves over the previous best bound of O(m^16 n^3 (log mn)^3) due to Dyer and Frieze

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron $$P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}$$ P = { x ∈ R n : A x ≤ b } , where $$A \in {\mathbb {Z}}^{m\times n}$$ A ∈ Z m × n . The bound is polynomial in $$n$$ n and the largest absolute value of a sub-determinant of $$A$$ A , denoted by $$\Delta$$ Δ . More precisely, we show that the diameter of $$P$$ P is bounded by $$O(\Delta ^2 n^4\log n\Delta )$$ O ( Δ 2 n 4 log n Δ ) . If $$P$$ P is bounded, then we show that the diameter of $$P$$ P is at most $$O(\Delta ^2 n^{3.5}\log n\Delta )$$ O ( Δ 2 n 3.5 log n Δ ) . For the special case in which $$A$$ A is a totally unimodular matrix, the bounds are $$O(n^4\log n)$$ O ( n 4 log n ) and $$O(n^{3.5}\log n)$$ O ( n 3.5 log n ) respectively. This improves over the previous best bound of $$O(m^{16}n^3(\log mn)^3)$$ O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1-16, 1994)

A Note on Non-Degenerate Integer Programs with Small Sub-Determinants

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the largest absolute value of an entry in $A$ and $m$ are constant. Then, this is applied to solve integer programs in inequality form in polynomial-time, where the absolute values of all maximal sub-determinants of $A$ lie between $1$ and a constant

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

FPT-algorithms for some problems related to integer programming

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.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author: some minor corrections has been don