Polynomial size IP formulations of knapsack may require exponentially large coefficients

Hojny, Christopher

doi:10.1016/j.orl.2020.07.013

Cited by 7 publications

(8 citation statements)

References 8 publications

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“…By Lemma 17, the list of observers Obs(X ) in counterclockwise order can be found Remark 19 One can show that the number |Y | of observers of X in Theorem 18 can be of order Θ(2 γ ), which means that the presented algorithm is not polynomial in the input size. However, if the points in V are encoded in unary, then the algorithm is indeed polynomial.…”

Section: Where γ Is An Upper Bound On the Binary Encoding Size Of Any Point In Vmentioning

confidence: 99%

“…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%

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Computational aspects of relaxation complexity: possibilities and limitations

2021

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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.

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Section: Where γ Is An Upper Bound On the Binary Encoding Size Of Any Point In Vmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computational aspects of relaxation complexity: possibilities and limitations

2021

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“…sboxes The test set comprises 18 instances modeling 4-bit (12 instances) and 5-bit (6 instances) S-boxes, which are certain non-sparse Boolean functions arising in symmetric-key cryptography. The derived sets X are contained in {0, 1} 8 and {0, 1} 10 , respectively, and Y are the complementary binary points. These instances have also been used by Udovenko [27] who solved the full model (3), i.e., without column generation.…”

Section: Test Setsmentioning

confidence: 99%

“…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

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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.

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“…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 [14]; see also [15] for a lower bound on the relative size of coefficients in a relaxation. For X ⊆ {0, 1} d , Jeroslow [16] 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 [13] and Taylor & Zwicker [27].…”

Section: Introductionmentioning

confidence: 99%

Computational Aspects of Relaxation Complexity: Possibilities and Limitations

Averkov¹,

Hojny²,

Schymura³

2021

Preprint

Self Cite

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The relaxation complexity 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 rcQ(X), a variant of rc(X) in which the polyhedra P are required to be rational, and we show that rc(X) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on rc(X) with the particular focus on the existence of a finite set Y ⊆ Z d such that separating X and Y \ X allows us to deduce rc(X) ≥ k. In particular, we show for some choices of X that no such finite set Y exists to certify the value of rc(X), providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for rc(X) for specific classes of sets X and present the first practically applicable approach to compute rc(X) for sets X that admit a finite certificate.

show abstract

Polynomial size IP formulations of knapsack may require exponentially large coefficients

Cited by 7 publications

References 8 publications

Computational aspects of relaxation complexity: possibilities and limitations

Computational aspects of relaxation complexity: possibilities and limitations

Efficient MIP Techniques for Computing the Relaxation Complexity

Computational Aspects of Relaxation Complexity: Possibilities and Limitations

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