A note on non-degenerate integer programs with small sub-determinants

Artmann, Stephan; Eisenbrand, Friedrich; Glanzer, Christoph; Oertel, Timm; Vempala, Santosh; Weismantel, Robert

doi:10.1016/j.orl.2016.07.004

Cited by 28 publications

(48 citation statements)

References 5 publications

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“…A consequence of Corollary 2 is that almost all problems can be solved in polynomial time provided ∆ is constant. This consequence can also be derived from a classic dynamic programming result by Gomory involving the so-called group relaxation [11] as well as from the dynamic programs presented in [1] or [9]. The running time of the latter has lesser dependence on ∆ than our approach in Corollary 2.…”

Section: The Value Pr(a) Can Be Interpreted As the Likelihood That Thmentioning

confidence: 69%

The Integrality Number of an Integer Program

Paat

Schlöter

Weismantel

2020

Lecture Notes in Computer Science

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We introduce the integrality number of an integer program (IP) in inequality form. Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor ∆ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers τ (∆) and κ(∆) such that an IP with n ≥ τ (∆) many variables and n + κ(∆) · √ n many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. A special instance of our results shows that IPs defined by only n constraints can be solved via a MIP relaxation with O( √ ∆) many integer constraints.1 Introduction.Let A ∈ Z m×n satisfy rank(A) = n and c ∈ Z n . We denote the integer linear program parameterized by right hand side b ∈ Z m by IP A,c (b) := max{c ⊺ x : Ax ≤ b and x ∈ Z n }.See [8] for more on parametric integer programs. We are interested in solving IP A,c (b) by relaxing it to have fewer integer constraints. In special cases we can solve IP A,c (b) with zero integer constraints by only solving its linear relaxation LP A,c (b) := max{c ⊺ x : Ax ≤ b}.However, these special cases require the underlying polyhedron to have optimal integral vertices, which occurs for instance when A is totally unimodular. Our target is to consider general matrices A and relaxations in the form of a mixed integer program:where k ∈ Z ≥0 and W ∈ Z k×n satisfies rank(W ) = k.One sufficient condition for solving IP A,c (b) using a mixed integer relaxation is that the vertices of W-MIP A,c (b) are integral. The vertices of W-MIP A,c (b) are the vertices of the polyhedron conv {x ∈ R n : Ax ≤ b, W x ∈ Z k } .

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Section: The Value Pr(a) Can Be Interpreted As the Likelihood That Thmentioning

confidence: 69%

The Integrality Number of an Integer Program

Paat

Schlöter

Weismantel

2020

Lecture Notes in Computer Science

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“…Let the matrix H have the additional property, such that H has no singular n× n submatrices. One result of [5] states that if n ≥ f (∆), then the matrix H has at most n + 1 rows, where f (∆) is a function that depends on ∆ only. The paper [5] contains a super-polynomial estimate on the value of f (∆).…”

Section: Some Auxiliary Resultsmentioning

confidence: 99%

“…The second aim of our paper is to improve results of [5]. Namely, in Section 4, we will present a FPT-algorithm for the ILPP ∆ , when the constraints matrix is close to a square matrix, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…This fact gives us a FPT-algorithm for the case, when the problem's constraints matrix has no singular rank submatrices. Indeed, such matrices can have only one additional row if the dimension is sufficiently large, due to [5]. In this paper, we present an algorithm with a better complexity bound.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we present an algorithm with a better complexity bound. Additionally, we improve some inequalities established in [5].…”

Section: Introductionmentioning

confidence: 99%

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FPT-algorithms for some problems related to integer programming

et al. 2018

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In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms with respect to the same parameter for the problems, when the matrices have no singular rank submatrices.

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