A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

Kaibel, Volker; Weltge, Stefan

doi:10.1007/s00454-014-9655-9

Cited by 48 publications

(34 citation statements)

References 15 publications

(15 reference statements)

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“…In particular, Theorem 1.2 immediately follows from the following result. We remark that this is a generalization of a result in [16], where the case t = 1 was established.…”

Section: Our Resultssupporting

confidence: 69%

Disjoint pairs in set systems with restricted intersection

Girão

Snyder

2020

European Journal of Combinatorics

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The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In particular, we show that for any pair of set systems (A, B) which avoid a cross-intersection of size t, the number of disjoint pairs (A, B) with A ∈ A and B ∈ B is at most t−1 k=0 n k 2 n−k . This implies an asymptotically best possible upper bound on the number of disjoint pairs in a single t-avoiding family F ⊂ P[n]. We also study this problem when A, B ⊂ [n] (r) are both r-uniform, and show that it is closely related to the problem of determining the maximum of the product |A||B| when A and B avoid a cross-intersection of size t, and n ≥ n 0 (r, t).

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“…In particular, Theorem 1.2 immediately follows from the following result. We remark that this is a generalization of a result in [16], where the case t = 1 was established.…”

Section: Our Resultssupporting

confidence: 69%

Disjoint pairs in set systems with restricted intersection

Girão

Snyder

2020

European Journal of Combinatorics

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“…The factor log(3/2) ≈ 0.585 in the exponent is the current best one due to [16]; for various approximate case versions see [3,5,7]. The first exponential lower bound was established in [12,13] by combining the seminal work of [20] together with an observation in [24].…”

Section: Unique Disjointnessmentioning

confidence: 98%

Average case polyhedral complexity of the maximum stable set problem

2016

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We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a $2^{\Omega(n/ \log n)}$ lower bound with probability at least $1 - 2^{-2^n}$ for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability $p \geq 2^{-\binom{n/4}{2}+n}$. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from $p = \Omega(\log^{6+\varepsilon} / n)$ to $p = o (1 / \log n)$. Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models

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“…Finally, since K h is a minor of K h,h , we have xc(COR(K h )) xc(COR(K h,h )). In [12], it is shown that xc(COR(K h )) (1.5) h . (A weaker exponential bound was given earlier in [9].)…”

Section: Proof It Is Easy To See That Hmentioning

confidence: 99%