Deterministic and randomized polynomial‐time approximation of radii

Brieden, Andreas; Gritzmann, Peter; Kannan, Ravindran; Klee, Victor; Lovász, László; Simonovits, Miklós

doi:10.1112/s0025579300014364

Cited by 35 publications

(44 citation statements)

References 39 publications

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“…(Bounds for the convex maximization problem that involve pth powers of norms have, of course, to be adjusted.) Let us point out that, as for polynomial-time approximations of B dr (p) (and in the appropriate model of computation), the given bound is tight for p ≥ 2; see [7] for more details and further results on the polynomial-time approximation of various other geometric functionals.…”

Section: Lemma 52mentioning

confidence: 97%

“…In fact, [6,7] give deterministic and randomized algorithms which in many cases are even asymptotically optimal for that task. With q again defined by for our polynomial approximations of the specific balls.…”

Section: Lemma 52mentioning

confidence: 98%

“…Of course, now we can again use the results of [7] to approximate B p,p by a polytope with polynomially many facets up to an error…”

Section: Lemma 52mentioning

confidence: 99%

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On Clustering Bodies: Geometry and Polyhedral Approximation

Brieden

Gritzmann

2009

Discrete Comput Geom

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The present paper studies certain classes of closed convex sets in finitedimensional real spaces that are motivated by their application to convex maximization problems, most notably, those evolving from geometric clustering. While these optimization problems are NP-hard in general, polynomial-time approximation algorithms can be devised whenever appropriate polyhedral approximations of their related clustering bodies are available. Here we give various structural results that lead to tight approximations.

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Section: Lemma 52mentioning

confidence: 97%

Section: Lemma 52mentioning

confidence: 98%

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On Clustering Bodies: Geometry and Polyhedral Approximation

Brieden

Gritzmann

2009

Discrete Comput Geom

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“…In [11] they applied (2) to prove that the so-called L 1 diameter of convex bodies can be approximated within a factor σ 1 (see Theorem 3.1 in [11]). By an inapproximate result of the L 1 diameter for convex bodies (see [3]), it follows that the ratio σ 1 cannot be improved in general. In fact, σ 1 is a tight bound not only for ξ ∼ B n but also for other distributions on the support set B n , also due to the inapproximate result of the L 1 diameters of convex bodies.…”

Section: Multilinear Tensor Function In Bernoulli Random Variablesmentioning

confidence: 99%

“…The inapproximate ratio of the L 2 diameter of convex bodies states that one cannot expect to find an approximation in any order better than Ω ln n n in polynomial-time unless P = N P (cf. [3]). …”

Section: Second We Havementioning

confidence: 99%

Probability Bounds for Polynomial Functions in Random Variables

Jiang

et al. 2014

Mathematics of OR

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Random sampling is a simple but powerful method in statistics and the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ R n . To do so, one may select a simpler (even finite) subset S 0 ⊆ S, randomly take some samples over S 0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S 0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξ i (1 ≤ i ≤ n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob f (ξ 1 , ξ 2 , · · · , ξ n ) ≥ c 1 n − d 2 F 1 ≥ c 2 , where c 1 , c 2 > 0 are two universal constants. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

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Cliques in High-Dimensional Random Geometric Graphs

Avrachenkov

Bobu

2019

Complex Networks and Their Applications VIII

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Random geometric graphs are good examples of random graphs with a tendency to demonstrate community structure. Vertices of such a graph are represented by points in Euclid space R d , and edge appearance depends on the distance between the points. Random geometric graphs were extensively explored and many of their basic properties are revealed. However, in the case of growing dimension d → ∞ practically nothing is known; this regime corresponds to the case of data with many features, a case commonly appearing in practice. In this paper, we focus on the cliques of these graphs in the situation when average vertex degree grows significantly slower than the number of vertices n with n → ∞ and d → ∞. We show that under these conditions random geometric graphs do not contain cliques of size 4 a.s. As for the size 3, we will present new bounds on the expected number of triangles in the case log 2 n d log 3 n that improve previously known results.

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Deterministic and randomized polynomial‐time approximation of radii

Cited by 35 publications

References 39 publications

On Clustering Bodies: Geometry and Polyhedral Approximation

On Clustering Bodies: Geometry and Polyhedral Approximation

Probability Bounds for Polynomial Functions in Random Variables

Cliques in High-Dimensional Random Geometric Graphs

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