Related papers: Block-structured Integer Programming: Can we Param…
We consider integer programming problems in standard form $\max \{c^Tx : Ax = b, \, x\geq 0, \, x \in Z^n\}$ where $A \in Z^{m \times n}$, $b \in Z^m$ and $c \in Z^n$. We show that such an integer program can be solved in time $(m…
The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the…
We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we…
Solving integer programs of the form $\min \{\mathbf{x} \mid A\mathbf{x} = \mathbf{b}, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \mathbf{x} \in \mathbb{Z}^n \}$ is, in general, $\mathsf{NP}$-hard. Hence, great effort has been put into…
It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $\Delta$ in absolute value, can be solved in polynomial time. We answer this question…
We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \,…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to…
Let $\Gamma$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(\Gamma,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$…
In the Integer Quadratic Programming problem input is an n*n integer matrix Q, an m*n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x in Z^n, minimizing x^TQx, subject to Ax <= b. We give a fixed…
There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that,…
We consider the following problem: Given a rational matrix $A \in \setQ^{m \times n}$ and a rational polyhedron $Q \subseteq\setR^{m+p}$, decide if for all vectors $b \in \setR^m$, for which there exists an integral $z \in \setZ^p$ such…
We study integer programming instances over polytopes P(A,b)={x:Ax<=b} where the constraint matrix A is random, i.e., its entries are i.i.d. Gaussian or, more generally, its rows are i.i.d. from a spherically symmetric distribution. The…
We study the parameterized complexity of algorithmic problems whose input is an integer set $A$ in terms of the doubling constant $C := |A + A|/|A|$, a fundamental measure of additive structure. We present evidence that this new…
We consider a family of integer linear programs in which the coefficients of the constraints and objective function are polynomials of an integer parameter $t.$ For $\ell$ in $\mathbb{Z}_+,$ we define $f_\ell(t)$ to be the…
The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{\Delta,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $\Delta$ and…
Let $n \ge 2$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) =…
We propose a new class of models for variable clustering called Asymptotic Independent block (AI-block) models, which defines population-level clusters based on the independence of the maxima of a multivariate stationary mixing random…
A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…
We determine the probability, structure dependent, that the block Wiedemann algorithm correctly computes leading invariant factors. This leads to a tight lower bound for the probability, structure independent. We show, using block size…