arXiv subjects

Dmitry Gribanov

Publications and source records attributed to Dmitry Gribanov.

6 recordsLinked to original sources

arXiv2026

Algorithms for Standard-form ILP Problems via Komlós' Discrepancy Setting (Refined $2^{O(k)}$-analysis)

We study the standard-form ILP problem $\max\{ c^\top x \colon A x = b,\; x \in Z_{\geq 0}^n \}$, where $A\in Z^{k\times n}$ has full row rank. We obtain refined FPT algorithms parameterized by $k$ and $Δ$, the maximum absolute value of a $k\times k$ minor of $A$. Our approach combines discrepancy-based dynamic programming with matrix $\ell_2$-discrepancy bounds in Komlós' setting. Up to polynomial factors in the input size, the optimization problem can be solved in time $2^{O(k)}Δ^2$, and the corresponding feasibility problem in time $2^{O(k)}Δ$. We would like to point out that the original version of this work, published in ESA 2026 proceedings, contains a slightly weaker result. More precisely, the earlier bounds were $O(κ_k)^{2k} Δ^2$ for optimization and $O(κ_k)^{k} Δ$ for feasibility, where $κ_k$ denotes the maximum $\ell_\infty$-discrepancy in Komlós' setting. However, following the advice of one of the anonymous reviewers -- to whom we are very grateful -- we replaced the $\ell_\infty$-discrepancy with the $\ell_2$-discrepancy in the analysis. This is what allowed us to replace dependencies of the form $κ_k^{O(k)}$ with $2^{O(k)}$.

cs.DS↗

arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗

arXiv2026

Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra

In our work, we consider the problem of computing a vector $x \in Z^n$ of minimum $\|\cdot\|_p$-norm such that $a^\top x \not= a_0$, for any vector $(a,a_0)$ from a given subset of $Z^n$ of size $m$. In other words, we search for a vector of minimum norm that avoids a given finite set of hyperplanes, which is natural to call as the $\textit{Hyperplanes Avoiding Problem}$. This problem naturally appears as a subproblem in Barvinok-type algorithms for counting integer points in polyhedra. We show that: 1) With respect to $\|\cdot\|_1$, the problem admits a feasible solution $x$ with $\|x\|_1 \leq (m+n)/2$, and show that such solution can be constructed by a deterministic polynomial-time algorithm with $O(n \cdot m)$ operations. Moreover, this inequality is the best possible. This is a significant improvement over the previous randomized algorithm, which computes $x$ with a guaranty $\|x\|_{1} \leq n \cdot m$. The original approach of A.~Barvinok can guarantee only $\|x\|_1 = O\bigl((n \cdot m)^n\bigr)$. To prove this result, we use a newly established algorithmic variant of the Combinatorial Nullstellensatz; 2) The problem is NP-hard with respect to any norm $\|\cdot\|_p$, for $p \in \bigl(R_{\geq 1} \cup \{\infty\}\bigr)$. 3) As an application, we show that the problem to count integer points in a polytope $P = \{x \in R^n \colon A x \leq b\}$, for given $A \in Z^{m \times n}$ and $b \in Q^m$, can be solved by an algorithm with $O\bigl(ν^2 \cdot n^3 \cdot Δ^3 \bigr)$ operations, where $ν$ is the maximum size of a normal fan triangulation of $P$, and $Δ$ is the maximum value of rank-order subdeterminants of $A$. As a further application, it provides a refined complexity bound for the counting problem in polyhedra of bounded codimension. For example, in the polyhedra of the Unbounded Subset-Sum problem.

cs.CC↗

arXiv2025

Diagonal Frobenius Number via Gomory's Relaxation and Discrepancy

For a matrix $A \in Z^{k \times n}$ of rank $k$, the diagonal Frobenius number $F_{\text{diag}}(A)$ is defined as the minimum $t \in Z_{\geq 1}$, such that, for any $b \in \text{span}_{Z}(A)$, the condition \begin{equation*} \exists x \in R_{\geq 0}^n,\, x \geq t \cdot 1 \colon \quad b = A x \end{equation*} implies that \begin{equation*} \exists z \in Z_{\geq 0}^n \colon\quad b = A z. \end{equation*} In this work, we show that \begin{equation*} F_{\text{diag}}(A) = Δ+ O(\log k), \end{equation*} where $Δ$ denotes the maximum absolute value of $k \times k$ sub-determinants of $A$. From the computational complexity perspective, we show that the integer vector $z$ can be found by a polynomial-time algorithm for some weaker values of $t$ in the described condition. For example, we can choose $t = O( Δ\cdot \log k)$ or $t = Δ+ O(\sqrt{k} \cdot \log k)$. Additionally, in the assumption that a $2^k$-time preprocessing is allowed or a base $J$ with $|{\det A_{J}}| = Δ$ is given, we can choose $t = Δ+ O(\log k)$. Finally, we define a more general notion of the diagonal Frobenius number for slacks $F_{\text{slack}}(A)$, which is a generalization of $F_{\text{diag}}(A)$ for canonical-form systems, like $A x \leq b$. All the proofs are mainly done with respect to $F_{\text{slack}}(A)$. The proof technique uses some properties of the Gomory's corner polyhedron relaxation and tools from discrepancy theory.

cs.DM↗

arXiv2024

Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.

cs.CC↗

arXiv2019

On integer programing with bounded determinants

Let $A$ be an $(m \times n)$ integral matrix, and let $P=\{ x : A x \leq b\}$ be an $n$-dimensional polytope. The width of $P$ is defined as $ w(P)=min\{ x\in \mathbb{Z}^n\setminus\{0\} :\: max_{x \in P} x^\top u - min_{x \in P} x^\top v \}$. Let $Δ(A)$ and $δ(A)$ denote the greatest and the smallest absolute values of a determinant among all $r(A) \times r(A)$ sub-matrices of $A$, where $r(A)$ is the rank of a matrix $A$. We prove that if every $r(A) \times r(A)$ sub-matrix of $A$ has a determinant equal to $\pm Δ(A)$ or $0$ and $w(P)\ge (Δ(A)-1)(n+1)$, then $P$ contains $n$ affine independent integer points. Also we have similar results for the case of \emph{$k$-modular} matrices. The matrix $A$ is called \emph{totally $k$-modular} if every square sub-matrix of $A$ has a determinant in the set $\{0,\, \pm k^r :\: r \in \mathbb{N} \}$. When $P$ is a simplex and $w(P)\ge δ(A)-1$, we describe a polynomial time algorithm for finding an integer point in $P$. Finally we show that if $A$ is \emph{almost unimodular}, then integer program $\max \{c^\top x :\: x \in P \cap \mathbb{Z}^n \}$ can be solved in polynomial time. The matrix $A$ is called \emph{almost unimodular} if $Δ(A) \leq 2$ and any $(r(A)-1)\times(r(A)-1)$ sub-matrix has a determinant from the set $\{0,\pm 1\}$.

cs.CG↗