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D. S. Malyshev

Publications and source records attributed to D. S. Malyshev.

9 recordsLinked to original sources
arXiv2024

Efficient search of a minimum tree on points in a space with the $l_1$-norm

In this paper, we consider the minimum spanning tree problem (for short, MSTP) on an arbitrary set of $n$ points of $d$-dimensional space in $l_1$-norm. For this problem, for each fixed $d\geq 2$, there is a known algorithm of the computational complexity $O\big(n\cdot (\log\,n + \log^{r_d}\,n\cdot \log\log\,n)\ big)$, where $r_d\in \{0,1,2,4\}$ for $d\in \{2,3,4,5\}$ and $r_d=d$ for $d\geq 6$. For $d=3$, this result can be improved to the computational complexity $O(n\cdot \log\,n)$. In this paper, for any fixed $d\geq 2$, an algorithm with the computational complexity $O(n\cdot \log^{d-1}\,n)$ is proposed to solve the considered MSTP, which improves the previous achievement for $d\geq 6$.

cs.DM↗
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↗
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↗