Distances to lattice points in knapsack polyhedra

Aliev, Iskander; Henk, Martin; Oertel, Timm

doi:10.1007/s10107-019-01392-1

Cited by 24 publications

(30 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Aliev et al [4,Theorem 1] proved that for any vertex x * of the polytope P (a, b) there exists an integer point z ∈ P (a, b) such that…”

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Aliev et al [4,Theorem 1] proved that for any vertex x * of the polytope P (a, b) there exists an integer point z ∈ P (a, b) such that…”

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.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This bound has been recently extended to the mixed-integer case by Paat et al [18] to p∆, where p is the number of integer variables. For other recent proximity results in Integer Linear Programming, we refer the reader to [11,26,1].…”

Section: Introductionmentioning

confidence: 99%

“…⋄ Example 1 explains the lack of proximity results in the nonconvex setting. However, a key observation is that the solution x * = t, while not optimal to (1), is 'almost' optimal. Furthermore, its distance from x c is always 3 4 .…”

Section: Introductionmentioning

confidence: 99%

Subdeterminants and Concave Integer Quadratic Programming

Pia¹

2019

SIAM J. Optim.

View full text Add to dashboard Cite

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of n∆ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and ∆ denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman and Magagnosc showed that the same upper bound is valid if a more general convex function is minimized, instead of a linear function. No proximity result of this type is known when the objective function is nonconvex. In fact, if we minimize a concave quadratic, no upper bound can be given as a function of n and ∆. Our key observation is that, in this setting, proximity phenomena still occur, but only if we consider also approximate solutions instead of optimal solutions only. In our main result we provide upper bounds on the distance between approximate (resp., optimal) solutions to a Concave Integer Quadratic Programming problem and optimal (resp., approximate) solutions of its continuous relaxation. Our bounds are functions of n, ∆, and a parameter ǫ that controls the quality of the approximation. Furthermore, we discuss how far from optimal are our proximity bounds.

show abstract

“…The motivation of this paper is to understand IP(b) by studying functions f whose input is IP(b), or equivalently, whose input is a vector b ∈ Z m . Such functions include the integrality gap function [4,17,26], the optimal value function [21,39], the running time of an algorithm as a function of b [3,32], and the flatness value [8,22]. Other examples include the sparsity function and the IP to LP distance function.…”

mentioning

confidence: 99%

“…It is not known if the bound in (1.4) is tight. In the case m = 1, Aliev et al [4] provide a tight upper bound on the related distance function…”

mentioning

confidence: 99%

The Distributions of Functions Related to Parametric Integer Optimization

Oertel¹,

Paat²,

Weismantel³

2020

SIAM J. Appl. Algebra Geometry

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Distances to lattice points in knapsack polyhedra

Cited by 24 publications

References 24 publications

Distance-Sparsity Transference for Vertices of Corner Polyhedra

Distance-Sparsity Transference for Vertices of Corner Polyhedra

Subdeterminants and Concave Integer Quadratic Programming

The Distributions of Functions Related to Parametric Integer Optimization

Contact Info

Product

Resources

About