Equal coefficients and tolerance in coloured tverberg partitions

Soberón, Pablo

doi:10.1145/2493132.2462369

Cited by 21 publications

(33 citation statements)

References 10 publications

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“…Recently, this was vastly improved by García-Colín, Raggi and Roldán-Pensado [19], who showed that for fixed r, d we have N(t, d, r) = rt + o(t). This settles the asymptotic behaviour of N(t, d, r) for large t, as the leading term matches the one for the lower bound N(t, d, r) rt + rd/2, first given in [33].…”

Section: Problem 13 (Tverberg Partitions With Tolerance) Given Posisupporting

confidence: 80%

“…We show a version of Tverberg's theorem with tolerance which holds for coloured classes. It should be noted that the adaptation of Sarkaria's methods described in [33] It is known that the probability p(r) that a random permutation of [r] has at least one fixed point tends to 1 − 1/e as r → ∞. Thus, the lemma above implies that, given at least one forbidden value for each element in [r], the probability that a random permutation of [r] hits at least one of them is at least p(r).…”

Section: Proof Let S Be a Set Of N Points Inmentioning

confidence: 99%

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Robust Tverberg and Colourful Carathéodory Results via Random Choice

Soberón¹

2017

Combinator. Probab. Comp.

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We use the probabilistic method to obtain versions of the colorful Carathéodory theorem and Tverberg's theorem with tolerance.In particular, we give bounds for the smallest integer N = N (t, d, r) such that for any N points in R d , there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.We prove a bound N = rt + O( √ t) for fixed r, d which is polynomial in each parameters. Our bounds extend to colorful versions of Tverberg's theorem, as well as Reay-type variations of this theorem. MSC2010 Classification: 52A35, 05D40

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Section: Problem 13 (Tverberg Partitions With Tolerance) Given Posisupporting

confidence: 80%

Section: Proof Let S Be a Set Of N Points Inmentioning

confidence: 99%

Robust Tverberg and Colourful Carathéodory Results via Random Choice

Soberón¹

2017

Combinator. Probab. Comp.

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“…Remark. Quite recently, Pablo Soberón [27] has found another (and simpler) proof of t(2, d) = 2. It starts with the observation that the vectors a i −b i , i ∈ [d+1] are linearly dependent, so ∑ d+1 1 γ i (a i −b i ) = 0 for some not all zero γ i .…”

Section: The Octahedral Constructionmentioning

confidence: 99%

“…A little extra is the efficient algorithm that follows from this proof. The paper [27] gives precise conditions for the existence of colourful partitions whose convex hulls have a common point with equal coefficients. The proof uses tensors as in Sarkaria's lemma which will be described in Section 7.…”

Section: The Octahedral Constructionmentioning

confidence: 99%

Tensors, colours, octahedra

Bárány¹

2014

Geometry, Structure and Randomness in Combinatorics

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“…More recently, Soberón [16] showed that if more color classes are available, then the conjecture holds for any k. More precisely, for P 1 , . .…”

Section: Introductionmentioning

confidence: 99%

No-dimensional Tverberg Theorems and Algorithms

Choudhary¹,

Mulzer²

2019

Preprint

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Tverberg's theorem is a classic result in discrete geometry. It states that for any integer k ≥ 2 and any finite d-dimensional point set P ⊂ R d of at least (d + 1)(k − 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The computational problem of finding such a partition lies in the complexity class PPAD ∩ PLS, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, i.e., sets in which each color appears exactly once, such that the convex hulls of the sets intersect. To date, the complexity of the corresponding computational problem has not been resolved.Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion. This allows it to relax the requirement on the cardinality of P . In fact, they prove a slightly stronger result that is based on the colorful Tverberg theorem. The argument is constructive, but it does not result in a polynomial-time algorithm.Here, we present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show that there is a deterministic algorithm that finds for any set P ⊂ R d of n points and any k ∈ {2, . . . , n} in O(nd log k ) time a partition of P into k subsets such that there is a ball of radius * Supported in part by ERC StG 757609.

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Equal coefficients and tolerance in coloured tverberg partitions

Cited by 21 publications

References 10 publications

Robust Tverberg and Colourful Carathéodory Results via Random Choice

Robust Tverberg and Colourful Carathéodory Results via Random Choice

Tensors, colours, octahedra

No-dimensional Tverberg Theorems and Algorithms

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