Optimizing Sparsity over Lattices and Semigroups

Aliev, Iskander; Averkov, Gennadiy; Loera, Jesús A. De; Oertel, Timm

doi:10.1007/978-3-030-45771-6_4

Cited by 12 publications

(25 citation statements)

References 23 publications

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“…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%

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Sparse representation of vectors in lattices and semigroups

et al. 2021

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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.

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

confidence: 86%

“…It is however not clear if it is possible to adapt this approach for the case of the 0 objective. This motivates the following problem: (1) and the functions ILR(A), ILC(A), ICR(A) and ICC(A) be computed in pseudo-polynomial time, i.e., when the input numbers are given in unary encoding rather than binary?…”

Section: Theoremmentioning

confidence: 99%

Sparse representation of vectors in lattices and semigroups

et al. 2021

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“…This is closely related to the results on the sparsity of systems of linear Diophantine equations in [2].…”

supporting

confidence: 59%

“…If c = 0 n , then σ(b) quantifies the sparsest feasible solution to IP(b). Upper bounds on σ(b) under this assumption were studied in [2,6]. Furthermore, Oertel et al [31] showed that asymptotic densities of σ can be bounded using the minimum absolute determinant of A or the 'number of prime factors' of the determinants.…”

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confidence: 99%

The Distributions of Functions Related to Parametric Integer Optimization

Oertel¹,

Paat²,

Weismantel³

2020

SIAM J. Appl. Algebra Geometry

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We create a framework for studying the asymptotic distributions of functions related to integer linear optimization. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand side is treated as input. We provide a spectrum of probability-like results that govern the overall asymptotic distribution of a function. We then apply this framework to the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. There has been a significant amount of research regarding the extreme values that these functions can attain. However, less is known about their typical values. Our results show that the typical values are smaller than the known worst case bounds.

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“…Bounds of this type are dated back to the classical integer Carathéodory theorems of Cook, Fonlupt and Schriver [10] and Sebő [17]. More recent contributions include results of Eisenbrand and Shmonin [12] and Aliev et al [1,2,3]. Further, in a very recent work Lee, Paat, Stallknecht and Xu [15] apply new sparsity-type bounds to refine the bounds for proximity.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distance-Sparsity Transference for Vertices of Corner Polyhedra

Aliev¹,

Celaya²,

Henk³

et al. 2021

SIAM J. Optim.

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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 implies that for any vertex x * of an integer feasible knapsack polytope P (a, b) = {x ∈ R n ≥0 : a ⊤ x = b}, a ∈ Z n >0 , there exists an integer point z * ∈ P (a, b) such that, denoting by s the size of the support of z * and assuming s > 0,where • ∞ stands for the ℓ∞-norm. The bound gives an exponential in s improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support.

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Optimizing Sparsity over Lattices and Semigroups

Cited by 12 publications

References 23 publications

Sparse representation of vectors in lattices and semigroups

Sparse representation of vectors in lattices and semigroups

The Distributions of Functions Related to Parametric Integer Optimization

Distance-Sparsity Transference for Vertices of Corner Polyhedra

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