Iskander Aliev

Cardiff University | CU · School of Mathematics

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solut...

We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it gives an exponential (in the size of support of a solution) improvement on previously known proximity estimates. In addition, for general integer...

We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric d-dimensional cube. This confirms a conjecture posed by Imre Bárány and Péter Frankl.

We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric d-dimensional cube. This confirms a conjecture posed by Imre Barany and Peter Frankl.

Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the -norm. Our main results are improved bounds on the -norm of sparse solutions to systems , where , and is either a g...

Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the $\ell_0$-norm. Our main results are improved bounds on the $\ell_0$-norm of sparse solutions to systems $A x = b$,...

We give an optimal upper bound for the \(\ell _{\infty }\)-distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack proble...

We construct a polynomial-time algorithm that, given a primitive positive integer vector a = (a_1, ..., a_n) and an integer b that satisfies a certain bound b >= G(a) + a_n, finds a nonnegative integer solution to the linear Diophantine equation a_1 x_1 + ... + a_n x_n = b. Here G(a) is the classical bound of Brauer for the Frobenius number associa...

We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and...

The support of a vector is the number of nonzero-components. We show that given an integral $m\times n$ matrix $A$, the integer linear optimization problem $\max\left\{\boldsymbol{c}^T\boldsymbol{x} : A\boldsymbol{x} = \boldsymbol{b}, \, \boldsymbol{x}\ge\boldsymbol{0}, \,\boldsymbol{x}\in\mathbb{Z}^n\right\}$ has an optimal solution whose support...

We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a “typical” knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario.

We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario.

Given an integral d × n matrix A, the well-studied affine semigroup \(\mathrm{Sg}(A) =\{ b: Ax = b,\ x \in \mathbb{Z}^{n},x \geq 0\}\) can be stratified by the number of lattice points inside the parametric polyhedra P A (b) = { x: Ax = b, x ≥ 0}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra,...

We give two structural results about the sparsity of solutions of the Diophantine system $Ax = b, x \geq 0, x \in \mathbb Z^t$.

Given a full-dimensional lattice Λ⊂Zk and a cost vector l∈Q>0k, we are concerned with the family of the group problems (0.1)min{l⋅x:x≡r(modΛ),x≥0},r∈Zk. The lattice programming gap gap(Λ,l) is the largest value of the minima in (0.1) as r varies over Zk. We show that computing the lattice programming gap is NP-hard when k is a part of input. We als...

The well-studied semigroup sg(A)={b: b=Ax, x non-negative integral vector} can be stratified by the sizes of the polyhedral fibers IP_A(b)={x: Ax=b, x non-negative integral vector}. The key theme of this paper is a structure theory that characterizes precisely the set sg_{\ge k}(A) of all vectors b in sg(A) such that their fiber IP_A(b) has at leas...

Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by "iterating" its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cut...

In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k,n), depending only...

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedr...

We obtain lower and upper bounds for the maximum gap of the group relaxations for integer programs.

We obtain lower and upper bounds for the maximum gap of the group relaxations for integer programs.

Given a matrix A∈ℤ m×n satisfying certain regularity assumptions, we consider for a positive integer s the set ℱ s (A)⊂ℤ m of all vectors b∈ℤ m such that the associated knapsack polytope P(A,b)={x∈ℝ ≥0 n :Ax=b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set ℱ...

We produce new upper and lower bounds for the s-Frobenius number by relating it to the so called s-covering radius of a certain convex body with respect to a certain lattice; this generalizes a well-known theorem of R. Kannan for the classical Frobenius number. Using these bounds, we obtain results on the average behavior of the s-Frobenius number,...

Given a primitive positive integer vector a, the Frobenius number F(a) is the largest integer that cannot be represented as a non-negative integral combination of the coordinates of a. We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bou...

Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point exists. We obtain a LLL-based polynomial time algorithm that solves the problem subject to a constraint on the l...

Given an integer mxn matrix A satisfying certain regularity assumptions, we consider the set F(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number associated with A. In this paper...

The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number.

Given a subvariety $V$ of the complex algebraic torus ${\mathbb G}_{\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\phi: {\mathbb G}_{\rm m}^n \to {\mathbb G}_{\rm m}^n$, the points ${\bf x}$ whose forward orbits ${\mathcal O}_\phi({\bf x})$ belong to $V$ form its {\em stable} subvariety $S(V,\phi)$. The main result o...

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an alg...

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an alg...

Let L(x)=a1x1+a2x2+⋅⋅⋅+anxn, n≥2, be a linear form with integer coefficients a1,a2,…,an which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a1,a2,…,an. The main result of this paper asserts that there exist linearl...

Let B be a Borel set in Ed with volume V(B) = ∞. It is shown that almost all lattices L in Ed contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in Ed with V(S ) = ∞. Then for almost all lattices L in Ed the successive minima λ1(...

Given N⩾2 positive integers a1,a2,…,aN with GCD(a1,…,aN)=1, let fN denote the largest natural number which is not a positive integer combination of a1,…,aN. This paper gives an optimal lower bound for fN in terms of the absolute inhomogeneous minimum of the standard (N−1)-simplex.

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals wit...

We study the problem of best approximations of a vector $\alpha\in{\mathbb R}^n$ by rational vectors of a lattice $\Lambda\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to...

Let L(x)=a 1x 1+a 2x 2+⋅⋅⋅+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that ther...

Estimates are given for the product of the lengths of integer vectors spanning a given linear subspace.