Colorful linear programming, Nash equilibrium, and pivots

Meunier, Frédéric; Sarrabezolles, Pauline

doi:10.48550/arxiv.1409.3436

Cited by 2 publications

(4 citation statements)

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“…Furthermore, we show that the global search variant of Ncp is NP-hard, which answers a question by Bárány and Onn [4]. This question was also answered independently by Meunier and Sarrabezolles [18].…”

Section: Introductionsupporting

confidence: 74%

“…This answers a question by Bárány and Onn [4, p. 561]. We note that this question has been answered independently by Meunier and Sarrabezolles [18,Theorem 2]. In contrast to the previous sections, all algorithms in this section are analyzed in the Word-Ram with logarithmic costs.…”

Section: The Complexity Of a Related Problemmentioning

confidence: 62%

“…In a recent result, Meunier et al [17] showed that ColorfulCarathéodory is contained in the intersection of two important subclasses of TFNP: polynomial parity argument in a directed graph (PPAD) and polynomial-time local search (PLS). Moreover, Meunier and Sarrabezolles [18] have shown that a related problem is PPADcomplete: given d + 1 pairs of points P 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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Computational Aspects of the Colorful Carathéodory Theorem

Mulzer

Stein

2018

Discrete Comput Geom

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Let C1, . . . , C d+1 ⊂ R d be d + 1 point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence p1, . . . , p d+1 with pi ∈ Ci, for i = 1, . . . , d + 1, a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory.We define a novel notion of approximation that is compatible with the polynomial-time reductions to Col-orfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed ε > 0, outputs an εd -colorful choice containing the origin in its convex hull in polynomial time.Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets C1, . . . , Cn ⊂ R d that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.

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Section: Introductionsupporting

confidence: 74%

Section: The Complexity Of a Related Problemmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

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Computational Aspects of the Colorful Carathéodory Theorem

Mulzer

Stein

2018

Discrete Comput Geom

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“…. , C d+1 that embraces the origin, it is PPAD-complete to find another colorful choice that embraces the origin [20]. Furthermore, given d + 1 point sets C 1 , .…”

Section: Introductionmentioning

confidence: 99%

The Rainbow at the End of the Line — A PPAD Formulation of the Colorful Carathéodory Theorem with Applications

Meunier¹,

Mulzer²,

Sarrabezolles³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Let C 1 , ..., C d+1 be d + 1 point sets in R d , each containing the origin in its convex hull. A subset C of d+1 i=1 C i is called a colorful choice (or rainbow) for C 1 , . . . , C d+1 , if it contains exactly one point from each set C i . The colorful Carathéodory theorem states that there always exists a colorful choice for C 1 , . . . , C d+1 that has the origin in its convex hull. This theorem is very general and can be used to prove several other existence theorems in high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg's theorem. The colorful Carathéodory problem (ColorfulCarathéodory) is the computational problem of finding such a colorful choice. Despite several efforts in the past, the computational complexity of ColorfulCarathéodory in arbitrary dimension is still open.We show that ColorfulCarathéodory lies in the intersection of the complexity classes PPAD and PLS. This makes it one of the few geometric problems in PPAD and PLS that are not known to be solvable in polynomial time. Moreover, it implies that the problem of computing centerpoints, computing Tverberg partitions, and computing points with large simplicial depth is contained in PPAD ∩ PLS. This is the first nontrivial upper bound on the complexity of these problems.Finally, we show that our PPAD formulation leads to a polynomial-time algorithm for a special case of ColorfulCarathéodory in which we have only two color classes C 1 and C 2 in d dimensions, each with the origin in its convex hull, and we would like to find a set with half the points from each color class that contains the origin in its convex hull.

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Colorful linear programming, Nash equilibrium, and pivots

Cited by 2 publications

References 0 publications

Computational Aspects of the Colorful Carathéodory Theorem

Computational Aspects of the Colorful Carathéodory Theorem

The Rainbow at the End of the Line — A PPAD Formulation of the Colorful Carathéodory Theorem with Applications

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