Colourful Linear Programming and its Relatives

Bárány, Imre; Onn, Shmuel

doi:10.1287/moor.22.3.550

Cited by 50 publications

(68 citation statements)

References 11 publications

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“…72, 2002 The Radon-split and the Helly-core of a point configuration 159 of this elegant theorem had been discovered in recent years, see [1,2,9] and references therein.…”

Section: Problem 2 Does Conv(s P (A)) = C P (A) Hold For All P and Ementioning

confidence: 99%

The Radon-split and the Helly-core of a point configuration

Onn¹

2001

Journal of Geometry

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“…72, 2002 The Radon-split and the Helly-core of a point configuration 159 of this elegant theorem had been discovered in recent years, see [1,2,9] and references therein.…”

Section: Problem 2 Does Conv(s P (A)) = C P (A) Hold For All P and Ementioning

confidence: 99%

The Radon-split and the Helly-core of a point configuration

Onn¹

2001

Journal of Geometry

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“…Algorithms for monochrome and colourful depth in higher dimension are an appealing challenge. Indeed, for (d + 1) colours in R d , it is not even clear how efficiently one can exhibit a single colourful simplex containing a given point [BO97], [DHST08].…”

Section: Conclusion and Questionsmentioning

confidence: 99%

Algorithms for Colourful Simplicial Depth and Medians in the Plane

Zasenko

Stephen

2016

Combinatorial Optimization and Applications

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Abstract. The colourful simplicial depth (CSD) of a point x ∈ R 2 relative to a configuration P = (P 1 , P 2 , . . . , P k ) of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point in R 2 , called a median, that maximizes colourful simplicial depth.For computing the colourful simplicial depth of x, our algorithm runs in time O (n log n + kn) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n 4 ). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O (n log n) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n 4 ) for finding a monochrome median.

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“…problems whose decision version has always a yes answer. The geometric algorithms introduced by Bárány [Bár82] and Bárány and Onn [BO97] and other methods to tackle the colourful feasibility problem, such as multi-update modifications, are studied and benchmarked in [DHST08]. The complexity of this challenging problem, i.e.…”

Section: Colourful Carathéodory Theoremsmentioning

confidence: 99%

A Further Generalization of the Colourful Carathéodory Theorem

Meunier

Deza

2013

Discrete Geometry and Optimization

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Given d +1 sets, or colours, S 1 , S 2 , . . . ,The convex hull of a colourful set S is called a colourful simplex. Bárány's colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of S i for i = 1, . . . , d + 1, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of S i ∪S j for 1 ≤ i < j ≤ d + 1 by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carathéodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of min i |S i | such simplices.

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Colourful Linear Programming and its Relatives

Cited by 50 publications

References 11 publications

The Radon-split and the Helly-core of a point configuration

The Radon-split and the Helly-core of a point configuration

Algorithms for Colourful Simplicial Depth and Medians in the Plane

A Further Generalization of the Colourful Carathéodory Theorem

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