Computational Aspects of the Colorful Carathéodory Theorem

Mulzer, Wolfgang; Stein, Yannik

doi:10.1007/s00454-018-9979-y

Cited by 7 publications

(9 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since we have no exact combinatorial polynomial-time algorithms for the colorful Carathéodory theorem, approximation iterative algorithms are of interest. This was first considered in [47], but other researchers, e.g., [285], have approached this problem too.…”

Section: Computational Considerationsmentioning

confidence: 99%

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

Loera

Goaoc

Meunier

et al. 2019

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Computational Considerationsmentioning

confidence: 99%

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

Loera

Goaoc

Meunier

et al. 2019

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In a celebrated result, the relevance of PPAD for algorithmic game theory was made clear when it turned out that computing a Nash-equilibrium in a two player game is PPAD-complete [CDT09]. In discrete geometry, finding a solution to the Colorful Carathéodory problem [Bár82] was shown to lie in the intersection PPAD ∩ PLS [MMSS17,MS18]. This further implies that finding a Tverberg partition (and computing a centerpoint) also lies in the intersection [Tve66,Sar92,LGMM19].…”

Section: Introductionmentioning

confidence: 99%

Computational Complexity of the $α$-Ham-Sandwich Problem

Chiu¹,

Choudhary²,

Mulzer³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

The classic Ham-Sandwich theorem states that for any d measurable sets in R d , there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and well-separated, then for any given α1, . . . , α d ∈ [0, 1], there is a unique oriented hyperplane that cuts off a respective fraction α1, . . . , α d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the α-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n(log n) d−3 ), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]).Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the α-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P -matrix linear complementarity problem.

show abstract

“…Since a Tverberg partition is guaranteed to exist if the cardinality of P is large enough, finding such a partition is a total search problem. In fact, the problem of computing a colorful Carathéodory traversal lies in the complexity class PPAD ∩ PLS [9,11], but no better upper bound on the difficulty of the problem is known. Since Sarkaria's proof can be interpreted as a polynomial-time reduction from the problem of finding a Tverberg partition to the problem of finding a colorful traversal, the same upper bound applies to finding Tverberg partitions.…”

Section: Introductionmentioning

confidence: 99%

No-dimensional Tverberg Theorems and Algorithms

Choudhary¹,

Mulzer²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Computational Aspects of the Colorful Carathéodory Theorem

Cited by 7 publications

References 31 publications

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

Computational Complexity of the $α$-Ham-Sandwich Problem

No-dimensional Tverberg Theorems and Algorithms

Contact Info

Product

Resources

About