Sparse Solutions of Linear Diophantine Equations

Aliev, Iskander; Loera, Jesús A. De; Oertel, Timm; O’Neill, Christopher

doi:10.1137/16m1083876

Cited by 35 publications

(48 citation statements)

References 27 publications

(35 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the knapsack case m = 1, the bound (5) strengthens Theorem 1.2 in [3] and, as it was already indicated in the IPCO version of this paper [1], it confirms a conjecture posed in [3, page 247]. Moreover, as a byproduct of the proof of Theorem 2 we obtain the following algorithmic result.…”

Section: Bounds For Icr(a) and Icc(a)supporting

confidence: 86%

Sparse representation of vectors in lattices and semigroups

et al. 2021

Self Cite

View full text Add to dashboard Cite

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 solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

show abstract

Section: Bounds For Icr(a) and Icc(a)supporting

confidence: 86%

Sparse representation of vectors in lattices and semigroups

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorem 1 (Upper bound on the discrete support function). There exists an optimal solution z * for Problem (1) such that…”

Section: Introductionmentioning

confidence: 99%

The Support of Integer Optimal Solutions

Aliev¹,

Loera²,

Eisenbrand³

et al. 2018

SIAM J. Optim.

View full text Add to dashboard Cite

The support of a vector is the number of nonzero-components. We show that given an integral m × n matrix A, the integer linear optimization problem max c T x : Ax = b, x ≥ 0, x ∈ Z n has an optimal solution whose support is bounded by 2m log(2 √ m A ∞ ), where A ∞ is the largest absolute value of an entry of A. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.

show abstract

“…The proof of Theorem 1 is based on a combination of group theory, lattice theory, and Ehrhart theory. On a high level, the combination of group and lattice theory bears similarities to papers of Gomory [11] and Aliev et al [2]. Gomory investigated the value function of an IP and proved its periodicity when the right-hand side vector is sufficiently large.…”

Section: Introductionmentioning

confidence: 67%

Sparsity of Integer Solutions in the Average Case

Oertel

Paat

Weismantel

2019

Integer Programming and Combinatorial Optimization

Self Cite

View full text Add to dashboard Cite

We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Carathéodory number provided that the determinants in the data are small.

show abstract

Sparse Solutions of Linear Diophantine Equations

Cited by 35 publications

References 27 publications

Sparse representation of vectors in lattices and semigroups

Sparse representation of vectors in lattices and semigroups

The Support of Integer Optimal Solutions

Sparsity of Integer Solutions in the Average Case

Contact Info

Product

Resources

About