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

On lattice point counting in Δ\Delta-modular polyhedra

Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating function for the power series $$\sum\limits_{m \in P \cap Z^n} x^m$$ can be computed with the arithmetic complexity $O\left(T_{SNF}(d) \cdot d^{k} \cdot d^{\log_2 \Delta}\right),$ where $k$ and $\Delta$ are fixed, $d = \dim P$, and $T_{SNF}(m)$ is the complexity to compute the Smith Normal Form for $m \times m$ integer matrix. In particular, $d = n$ for the case (i) and $d = n-k$ for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. We apply these results to parametric polytopes, and show that the step polynomial representation of the function $c_P(y) = |P_{y} \cap Z^n|$, where $P_{y}$ is parametric polytope, can be computed by a polynomial time even in varying dimension if $P_{y}$ has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients $e_i(P,m)$ of the Ehrhart quasi-polynomial $$\left| mP \cap Z^n\right| = \sum\limits_{j = 0}^n e_i(P,m)m^j$$ can be computed by a polynomial time algorithm for fixed $k$ and $\Delta$