arXiv subjects

I. A. Shumilov

Publications and source records attributed to I. A. Shumilov.

3 recordsLinked to original sources

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↗