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

Loera, Jesús A. De; Goaoc, Xavier; Meunier, Frédéric; Mustafa, Nabil H.

doi:10.1090/bull/1653

Cited by 66 publications

(34 citation statements)

References 395 publications

(472 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As it happens, the PPA-completeness of Discrete Ham Sandwich follows straightforwardly, and we present that next. The basic idea of Theorem 2.3 of embedding the necklace in the moment curve appears already in [62,51] and [48], p.48. Proof.…”

Section: Problems and Resultsmentioning

confidence: 99%

The complexity of splitting necklaces and bisecting ham sandwiches

Filos-Ratsikas

Goldberg

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

We resolve the computational complexity of two problems known as Necklace-splitting and Discrete Ham Sandwich showing that they are PPA-complete. For Necklace-splitting, this result is specific to the important special case in which two thieves share the necklace. We do this via a PPA-completeness result for an approximate version of the Consensus-halving problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional Möbius strip in the Consensus-halving problem. These results settle the status of PPA as a class that captures the complexity of "natural" problems whose definitions do not incorporate a circuit.Definition 1 An instance of the problem Leaf consists of an undirected graph G whose vertices have degree at most 2; G has 2 n vertices represented by bitstrings of length n; G is presented concisely via a circuit that takes as input a vertex and outputs its neighbour(s). We stipulate that vertex 0 n has degree 1. The challenge is to find some other vertex having degree 1.Complete problems for the class PPA seemed to be much more elusive than PPAD-complete ones, especially when one is interested in "natural" problems, where "natural" here has the very specific meaning of problems that do not explicitly contain a circuit in their definition. Besides Papadimitriou [60], other papers asking about the possible existence of natural PPA-complete problems include [36,13,19,22]. In a recent precursor [29] to the present paper we identified the first example of such a problem, namely the approximate Consensus-halving problem, dispelling the suspicion that such problems might not exist. In this paper we build on that result and settle the complexity of two natural and important problems whose complexity status were raised explicitly as open problems in Papadimitriou's paper itself, and in many other papers beginning in the 1980s. Specifically, we prove that Necklace-splitting (with two thieves, see Definition 2) and Discrete Ham Sandwich are both PPA-complete.Definition 2 (Necklace Splitting) In the k-Necklace-splitting problem there is an open necklace with ka i beads of colour i, for 1 ≤ i ≤ n. An "open necklace" means that the beads form a string, not a cycle. The task is to cut the necklace in (k − 1) · n places and partition the resulting substrings into k collections, each containing precisely a i beads of colour i, 1 ≤ i ≤ n.In Definition 2, k is thought of as the number of thieves who desire to split the necklace in such a way that the beads of each colour are equally shared. In this paper, usually we have k = 2 and we refer to this special case as Necklace-splitting.Definition 3 (Discrete Ham Sandwich) In the Discrete Ham Sandwich problem, there are n sets of points in n dimensions having integer coordinates (equivalently one could use rationals). A solution consists of a hyperplane that splits each set of points into subsets of equal size (if any points lie on the plane, we are allowed ...

show abstract

Section: Problems and Resultsmentioning

confidence: 99%

The complexity of splitting necklaces and bisecting ham sandwiches

Filos-Ratsikas

Goldberg

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…The purpose of this survey is to give a broad overview of the current state of the field and point out key open problems. Other surveys covering different aspects of Tverberg's theorem can be found in [Eck79,Eck93,Mat02,BBZ16,DLGMM17,BZ17].…”

Section: Introductionmentioning

confidence: 99%

Tverberg’s theorem is 50 years old: A survey

Bárány

Soberón

2018

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány

show abstract

“…In particular, our paper contributes to the current vibrant research activity focused on variations and generalizations of Sperner's lemma. Examples, with many references, are discussed in a survey by De Loera et al [5].…”

Section: Introductionmentioning

confidence: 99%

Envy-free cake division without assuming the players prefer nonempty pieces

Meunier

Zerbib

2019

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.

show abstract

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

Cited by 66 publications

References 395 publications

The complexity of splitting necklaces and bisecting ham sandwiches

The complexity of splitting necklaces and bisecting ham sandwiches

Tverberg’s theorem is 50 years old: A survey

Envy-free cake division without assuming the players prefer nonempty pieces

Contact Info

Product

Resources

About