Sparsity of Integer Solutions in the Average Case

Oertel, Timm; Paat, Joseph; Weismantel, Robert

doi:10.1007/978-3-030-17953-3_26

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…For other results regarding sparsity, see [3,10,19] for general A and [5,6,7,22] for matrices that form a Hilbert basis. See also the manuscript of Aliev et al [1], who give improved sparsity bounds for feasible solutions to special classes of integer programs and provide efficient algorithms for finding such solutions.…”

Section: Aliev Et Al [2] Established Thatmentioning

confidence: 99%

Improving proximity bounds using sparsity

Lee

Paat

Stallknecht

et al. 2020

Preprint

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We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

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Section: Aliev Et Al [2] Established Thatmentioning

confidence: 99%

Improving proximity bounds using sparsity

Lee

Paat

Stallknecht

et al. 2020

Preprint

Self Cite

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show abstract

“…Sparsity of solutions to linear Diophantine equations is relevant for the theory of compressed sensing for integer-valued signals [17,18,24], motivated by many applications in which the signal is known to have integer entries, for instance, in wireless communication [31] and in the theory of error-correcting codes [10]. Support minimization was also investigated in connection to integer optimization [2,16,29,30]. Also, numerous applications to combinatorial optimization problems have been explored.…”

Section: Introductionmentioning

confidence: 99%

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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“…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. If, in addition, A has the Hilbert basis property (i.e., if the columns of A correspond to a Hilbert basis of the cone generated by A), then bounds on σ(b) can be given solely in terms of m. Cook et al [15] showed that if σ(b) < ∞, then σ(b) ≤ 2m − 1; this was improved to σ(b) ≤ 2m − 2 by Sebő [36].…”

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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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Sparsity of Integer Solutions in the Average Case

Cited by 8 publications

References 10 publications

Improving proximity bounds using sparsity

Improving proximity bounds using sparsity

Sparse representation of vectors in lattices and semigroups

The Distributions of Functions Related to Parametric Integer Optimization

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