On the existence of compact<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>ε</mml:mi></mml:math>-approximated formulations for knapsack in the original space

Faenza, Yuri; Sanità, Laura

doi:10.1016/j.orl.2015.04.004

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…Although we will not rely on this interpretation in what follows, the idea behind our lower bound is based on the observation that the dimension of a simple game W can be seen as the chromatic number of a particular hypergraph H: the nodes of H are the losing coalitions, and a set of losing coalitions N forms a hyperedge iff N ∩ W ′ = ∅ for every weighted game W ′ ⊇ W. The proof of Kurz and Napel [7] establishes that H contains a clique of cardinality 7, which directly implies that the chromatic number of H is at least 7. This idea has been used previously in the context of lower bounds on sizes of integer programming formulations [4,5,3]. While we have not found any simple subgraph of larger chromatic number, we will show that H contains a hypergraph on 15 nodes whose chromatic number is 8.…”

Section: Introductionmentioning

confidence: 60%

Improved lower bound on the dimension of the EU council's voting rules

Kober¹,

Weltge²

2020

Preprint

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Kurz and Napel (2015) proved that the voting system of the EU council (based on the 2014 population data) cannot be represented as the intersection of six weighted games, i.e., its dimension is at least 7. This set a new record for real-world voting rules and the authors posed the exact determination as a challenge. Recently, Chen, Cheung, and Ng (2019) showed that the dimension is at most 24.We provide the first improved lower bound and show that the dimension is at least 8.

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Section: Introductionmentioning

confidence: 60%

Improved lower bound on the dimension of the EU council's voting rules

Kober¹,

Weltge²

2020

Preprint

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show abstract

“…Although we will not rely on this interpretation in what follows, the idea behind our lower bound is based on the observation that the dimension of a simple game W can be seen as the chromatic number of a particular hypergraph H : the nodes of H are the losing coalitions, and a set of losing coalitions N forms a hyperedge iff N ∩ W = ∅ for every weighted game W ⊇ W. The proof of Kurz and Napel [11] establishes that H contains a clique of cardinality 7, which directly implies that the chromatic number of H is at least 7. This idea has been used previously in the context of lower bounds on sizes of integer programming formulations [4,8,9]. While we have not found any simple subgraph of larger chromatic number, we will show that H contains a hypergraph on 15 nodes whose chromatic number is 8.…”

Section: Introductionmentioning

confidence: 66%

Improved lower bound on the dimension of the EU council’s voting rules

Kober

Weltge

2020

Optim Lett

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proved that the voting system of the EU council (based on the 2014 population data) cannot be represented as the intersection of six weighted games, i.e., its dimension is at least 7. This set a new record for real-world voting rules and the authors posed the exact determination as a challenge. Recently, Chen et al. (An upper bound on the dimension of the voting system of the European Union Council under the Lisbon rules, 2019, arXiv:1907.09711) showed that the dimension is at most 24. We provide the first improved lower bound and show that the dimension is at least 8.

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“…Proof. Assume that F is given as in (7). Since D Cy (d, d 2 ) and Q Cy (d, d 2 ) are d-polytopes by Remarks 9 and 10, by Remark 6 we have that…”

Section: The Polar Of a Cyclic Polytopementioning

confidence: 99%

“…Hence by Lemma 16, the number of inequalities needed to describe an ε-approximation of Q △ d is at least |S|/(dt). (Our proof approach can be interpreted as an extension of those in [7,16] where we used the well-known bound k j=0 n j ≤ 2 nH(k/n) that is valid for k ≤ n 2 and uses the entropy function H(p) = −x log 2 (p) − (1 − p) log 2 (1 − p) (see e.g. [12]).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Balas formulation for the union of polytopes is optimal

2019

Self Cite

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A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled.In this paper we show that this is best possible: in every dimension there exist two nonempty polytopes such that if a formulation for the convex hull of their union has a number of inequalities that is polynomial in the number of inequalities that describe the two polytopes, then the number of additional variables is at least linear in the dimension of the polytopes.We then show that this result essentially carries over if one wants to approximate the convex hull of the union of two polytopes and also in the more restrictive setting of lift-and-project.

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

Cited by 5 publications

References 17 publications

Improved lower bound on the dimension of the EU council's voting rules

Improved lower bound on the dimension of the EU council's voting rules

Improved lower bound on the dimension of the EU council’s voting rules

Balas formulation for the union of polytopes is optimal

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