Strong IP formulations need large coefficients

Hojny, Christopher

doi:10.1016/j.disopt.2021.100624

Cited by 7 publications

(7 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [3], this lower bound has been improved and computability also for d = 3 has been established. The interplay between the number of inequalities in a relaxation and the size of their coefficients has been investigated in [19]; see also [20] for a lower bound on the relative size of coefficients in a relaxation. For X ⊆ {0, 1} d , Jeroslow [21] derived an upper bound on rc(X , {0, 1} d ), which is an important subject in the area of social choice, see, e.g., Hammer et al [18] and Taylor and Zwicker [33].…”

Section: Introductionmentioning

confidence: 99%

Computational aspects of relaxation complexity: possibilities and limitations

2021

Self Cite

View full text Add to dashboard Cite

The relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is a useful tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on $${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$ rc Q ( X ) , a variant of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) in which the polyhedra P are required to be rational, and we show that $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) with the particular focus on the existence of a finite set $$Y \subseteq \mathbb {Z}^d$$ Y ⊆ Z d such that separating X and $$Y \setminus X$$ Y \ X allows us to deduce $${{\,\mathrm{rc}\,}}(X) \ge k$$ rc ( X ) ≥ k . In particular, we show for some choices of X that no such finite set Y exists to certify the value of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) , providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for specific classes of sets X and present the first practically applicable approach to compute $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for sets X that admit a finite certificate.

show abstract

Section: Introductionmentioning

confidence: 99%

Computational aspects of relaxation complexity: possibilities and limitations

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…In our notation this number equals rc(X, {0, 1} d ), and in yet a different language, Jeroslow [20] proved that rc(X, {0, 1} d ) ≤ 2 d−1 and exhibited examples that attain equality. Hojny [18] studied relaxations within {0, 1} d with respect to the size of the coefficients used.…”

Section: Separation Problemmentioning

confidence: 99%

“…However, since tight descriptions can have a very large size, we believe that one should not only focus on the tightness but also investigate possibilities of finding formulations of small size. While the size of the description is determined by the number of constraints and the size of the coefficients, here we only address problems related to the number of constraints (regarding the size of the coefficients, we refer to the recent work [18] of Hojny).…”

Section: Introductionmentioning

confidence: 99%

Complexity of linear relaxations in integer programming

Averkov¹,

Schymura²

2020

Preprint

View full text Add to dashboard Cite

For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity rc(X). This parameter was introduced by Kaibel & Weltge (2015) and captures the complexity of linear descriptions of X without using auxiliary variables.Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding rc(X) and its variant rc Q (X), restricting the descriptions of X to rational polyhedra.As our main results we show that rc(X) = rc Q (X) when: (a) X is at most four-dimensional, (b) X represents every residue class in (Z/2Z) d , (c) the convex hull of X contains an interior integer point, or (d) the lattice-width of X is above a certain threshold. Additionally, rc(X) can be algorithmically computed when X is at most three-dimensional, or X satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on rc(X) in terms of the dimension of X.

show abstract

“…In particular, if ε approaches 0, then rc ε (X) converges towards rc Q (X), a variant of the relaxation complexity which requires the relaxations to be rational. Further variations of rc(X) in which the size of coefficients in facet defining inequalities are bounded are discussed in [10,11].…”

Section: Introductionmentioning

confidence: 99%

Efficient MIP Techniques for Computing the Relaxation Complexity

Averkov¹,

Hojny²,

Schymura³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning.We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously.

show abstract

Strong IP formulations need large coefficients

Cited by 7 publications

References 31 publications

Computational aspects of relaxation complexity: possibilities and limitations

Computational aspects of relaxation complexity: possibilities and limitations

Complexity of linear relaxations in integer programming

Efficient MIP Techniques for Computing the Relaxation Complexity

Contact Info

Product

Resources

About