arXiv subjects

D. V. Gribanov

Publications and source records attributed to D. V. Gribanov.

12 recordsLinked to original sources

arXiv2023

A faster algorithm for counting the integer points number in $Δ$-modular polyhedra (corrected version)

Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $Δ$-modular, where the polytope $P$ is called $Δ$-modular if all the rank sub-determinants of $A$ are bounded by $Δ$ in the absolute value. We present a new FPT-algorithm, parameterized by $Δ$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $Δ$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $Δ$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $Δ$-modular subset-sum problem.

cs.CC↗

arXiv2022

On a Simple Connection Between $Δ$-modular ILP and LP, and a New Bound on the Number of Integer Vertices

Let $A \in Z^{m \times n}$, $rank(A) = n$, $b \in Z^m$, and $P$ be an $n$-dimensional polyhedron, induced by the system $A x \leq b$. It is a known fact that if $F$ is a $k$-face of $P$, then there exist at least $n-k$ linearly independent inequalities of the system $A x \leq b$ that become equalities on $F$. In other words, there exists a set of indices $J$, such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and $$ A_{J} x - b_{J} = 0,\quad \text{for any $x \in F$}. $$ We show that a similar fact holds for the integer polyhedron $$ P_{I} = conv.hull\bigl(P \cap Z^n\bigr), $$ if we additionally suppose that $P$ is $Δ$-modular, for some $Δ\in \{1,2,\dots\}$. More precisely, if $F$ is a $k$-face of $P_{I}$, then there exists a set of indices $J$, such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and $$ A_{J} x - b_{J} \oversetΔ{=} 0,\quad \text{for any $x \in F \cap Z^n$}, $$ where $x \oversetΔ{=} y$ means that $\|x - y\|_{\infty} < Δ$. In other words, there exist at least $n-k$ linearly independent inequalities of the system $A x \leq b$ that almost become equalities on $F \cap Z^n$. When we say almost, we mean that the slacks are not greater than $Δ-1$. Using this fact, we prove the inequality $$ |vert(P_I)| \leq 2 \cdot \binom{m}{n} \cdot Δ^{n-1}, $$ for the number of vertices of $P_I$, which is better, than the state of the art bound for $Δ= O(n^2)$.

cs.DM↗

arXiv2022

Structured $(\min,+)$-Convolution And Its Applications For The Shortest Vector, Closest Vector, and Separable Nonlinear Knapsack Problems

In this work we consider the problem of computing the $(\min, +)$-convolution of two sequences $a$ and $b$ of lengths $n$ and $m$, respectively, where $n \geq m$. We assume that $a$ is arbitrary, but $b_i = f(i)$, where $f(x) \colon [0,m) \to \mathbb{R}$ is a function with one of the following properties: 1. the linear case, when $f(x) =β+ α\cdot x$; 2. the monotone case, when $f(i+1) \geq f(i)$, for any $i$; 3. the convex case, when $f(i+1) - f(i) \geq f(i) - f(i-1)$, for any $i$; 4. the concave case, when $f(i+1) - f(i) \leq f(i) - f(i-1)$, for any $i$; 5. the piece-wise linear case, when $f(x)$ consist of $p$ linear pieces; 6. the polynomial case, when $f \in \mathbb{Z}^d[x]$, for some fixed $d$. To the best of our knowledge, the cases 4-6 were not considered in literature before. We develop true sub-quadratic algorithms for them. We apply our results to the knapsack problem with a separable nonlinear objective function, shortest lattice vector, and closest lattice vector problems.

cs.CC↗

arXiv2022

On $Δ$-Modular Integer Linear Problems In The Canonical Form And Equivalent Problems

Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $\max\{c^\top x \colon A x = b,\, x \in \mathbb{Z}^n_{\geq 0}\}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $\|A\|_{\max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word "bounded" if $x \leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$\max\{c^\top x \colon b_l \leq A x \leq b_r,\, x \in \mathbb{Z}^n\},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $Δ(A)$, where $Δ(A)$ is the maximum of $n \times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $Δ(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m \in \{0,1\}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.

cs.CC↗

arXiv2021

An FPTAS for the $Δ$-modular multidimensional knapsack problem

It is known that there is no EPTAS for the $m$-dimensional knapsack problem unless $W[1] = FPT$. It is true already for the case, when $m = 2$. But, an FPTAS still can exist for some other particular cases of the problem. In this note, we show that the $m$-dimensional knapsack problem with a $Δ$-modular constraints matrix admits an FPTAS, whose complexity bound depends on $Δ$ linearly. More precisely, the proposed algorithm complexity is $$O(T_{LP} \cdot (1/\varepsilon)^{m+3} \cdot (2m)^{2m + 6} \cdot Δ),$$ where $T_{LP}$ is the linear programming complexity bound. In particular, for fixed $m$ the arithmetical complexity bound becomes $$ O(n \cdot (1/\varepsilon)^{m+3} \cdot Δ). $$ Our algorithm is actually a generalisation of the classical FPTAS for the $1$-dimensional case. Strictly speaking, the considered problem can be solved by an exact polynomial-time algorithm, when $m$ is fixed and $Δ$ grows as a polynomial on $n$. This fact can be observed combining previously known results. In this paper, we give a slightly more accurate analysis to present an exact algorithm with the complexity bound $$ O(n \cdot Δ^{m + 1}), \quad \text{ for $m$ being fixed}. $$ Note that the last bound is non-linear by $Δ$ with respect to the given FPTAS.

cs.CC↗

arXiv2021

On lattice point counting in $Δ$-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 $Δ$ 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 Δ}\right), $ where $k$ and $Δ$ 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 $Δ$.

cs.CC↗

arXiv2020

A polynomial algorithm for minimizing discrete convic functions in fixed dimension

Recently classes of conic and discrete conic functions were introduced. In this paper we use the term convic instead conic. The class of convic functions properly includes the classes of convex functions, strictly quasiconvex functions and the class of quasiconvex polynomials. On the other hand, the class of convic functions is properly included in the class of quasiconvex functions. The discrete convic function is a discrete analogue of the convic function. Recently the lower bound $3^{n-1}\log (2 ρ-1)$ for the number of calls to the comparison oracle needed to find the minimum of the discrete convic function defined on integer points of some $n$-dimensional ball with radius $ρ$ was obtained. But the problem of the existence of a polynomial (in $\logρ$ for fixed $n$) algorithm for minimizing such functions has remained open. In this paper, we answer positively the question of the existence of such an algorithm. Namely, we propose an algorithm for minimizing discrete convic functions that uses $2^{O(n^2 \log n)} \log ρ$ calls to the comparison oracle and has $2^{O(n^2 \log n)} \mbox{poly }(\log ρ)$ bit complexity.

math.OC↗

arXiv2020

On the Proximity of the Optimal Values of the Multi-Dimensional Knapsack Problem with and without the Cardinality Constraint

We study the proximity of the optimal value of the m-dimensional knapsack problem to the optimal value of that problem with the additional restriction that only one type of items is allowed to include in the solution. We derive exact and asymptotic formulas for the precision of such approximation, i.e. for the infinum of the ratio of the optimal value for the objective functions of the problem with the cardinality constraint and without it. In particular, we prove that the precision tends to 0.59136.../m if n tends to infinity and m is fixed. Also, we give the class of the worst multi-dimensional knapsack problems for which the bound is attained. Previously, similar results were known only for the case m=1.

math.OC↗

arXiv2019

On the complexity of quasiconvex integer minimization problem

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

math.OC↗

arXiv2019

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.

math.OC↗

arXiv2017

FPT-algorithms for The Shortest Lattice Vector and Integer Linear Programming Problems

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 FPT-algorithms with respect to the same main parameter for the problems, when the matrices have no singular rank sub-matrices.

math.OC↗

arXiv2017

The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices

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 integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.

math.OC↗