Lower bounds on the sizes of integer programs without additional variables

Kaibel, Volker; Weltge, Stefan

doi:10.1007/s10107-014-0855-0

Cited by 27 publications

(56 citation statements)

References 13 publications

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“…As ε was arbitrary and the choices of n were infinite, this implies Theorem 1. We remark that similar strategies have been recently used by Kaibel and Weltge [7] and Kolliopoulos and Moysoglou [10].…”

Section: No Psrs In the Original Space Exists For Knapsackmentioning

confidence: 87%

On the existence of compactε-approximated formulations for knapsack in the original space

Faenza

Sanità

2015

Operations Research Letters

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Section: No Psrs In the Original Space Exists For Knapsackmentioning

confidence: 87%

On the existence of compactε-approximated formulations for knapsack in the original space

Faenza

Sanità

2015

Operations Research Letters

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“…The second theorem, Theorem 3.14, implies that for X ⊆ Z n finite, c(X) ≤ rc(X)(2 n − 1), where rc(X) denotes the minimum number of inequalities in any system whose integer solutions are exactly X, or −∞ if there is no such system. The quantity rc(X), known as the relaxation complexity of X, was studied in [26]. A similar quantity was studied earlier in [25].…”

Section: Tighter Bounds For Specified Arrangementsmentioning

confidence: 98%

Sublinear Bounds for a Quantitative Doignon--Bell--Scarf Theorem

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Hildebrand²,

Zenklusen

2018

SIAM J. Discrete Math.

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The recent paper "A quantitative Doignon-Bell-Scarf Theorem" by Aliev et al. generalizes the famous Doignon-Bell-Scarf Theorem on the existence of integer solutions to systems of linear inequalities. Their generalization examines the number of facets of a polyhedron that contains exactly k integer points in R n . They show that there exists a number c(n, k) such that any polyhedron in R n that contains exactly k integer points has a relaxation to at most c(n, k) of its inequalities that will define a new polyhedron with the same integer points. They prove that c(n, k) = O(k)2 n . In this paper, we improve the bound asymptotically to be sublinear in k, that is, c(n, k) = o(k)2 n . We also provide lower bounds on c(n, k), along with other structural results. For dimension n = 2, our upper and lower bounds match to within a constant factor.

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“…Despite the fact that we do not believe that all stable set polytopes admit polynomial-size MILEFs with o(n) integer variables, another motivation for improving the lower bound is the following. In [20,Prop. 2] it is mentioned that if a family of polytopes P with vertices in {0, 1} d admits a polynomial-time algorithm to decide whether a point in {0, 1} d belongs to P , then P can be described by a MILEF whose size is polynomial in d and that uses only d integer variables.…”

Section: Towards Tight Boundsmentioning

confidence: 99%

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Cevallos

Weltge

Zenklusen

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size.We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, travelling salesman polytope or dominant of the odd-cut polytope, needs Ω(n/ log n) many integer variables, where n is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only O(n) or O(n log n) (for the traveling salesman polytope) many integer variables.Our results build upon a new decomposition technique that, for any convex set C, allows for approximating any mixed-integer description of C by the intersection of C with the union of a small number of affine subspaces.

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Lower bounds on the sizes of integer programs without additional variables

Cited by 27 publications

References 13 publications

On the existence of compactε-approximated formulations for knapsack in the original space

On the existence of compactε-approximated formulations for knapsack in the original space

Sublinear Bounds for a Quantitative Doignon--Bell--Scarf Theorem

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

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