Transversal numbers for hypergraphs arising in geometry

Alon, Noga; Kalai, Gil; Matouek, Ji; Meshulam, Roy

doi:10.1016/s0196-8858(02)00003-9

Cited by 81 publications

(114 citation statements)

References 19 publications

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“…´ Ä µ as in Section 2.1. We draw in the plane using the same drawing rule described in Section 2.1, 5 We partition into two subsets ½ ¾ of size at most Ò ¾ each so…”

Section: ¾ 4 Improving the Tamaki-tokuyama Boundmentioning

confidence: 99%

Lenses in arrangements of pseudo-circles and their applications

Agarwal

Nevo

Pach

et al. 2004

J. ACM

161

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A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an empty lens if it does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of Ò pseudo-circles with the property that any two curves intersect precisely twice. This bound implies that any collection of Ò Ü -monotone pseudo-circles can be cut into Ç´Ò µ arcs so that any two intersect at most once; this improves a previous bound of Ç´Ò ¿ µ due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to Ç´Ò ¿ ¾´Ð Ó Òµ Ç´« ×´Ò µµ µ, where «´Òµ is the inverse Ackermann function, and × is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudocircles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to Ç´Ò ¿ µ. As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudo-circles, of arbitrary Ü-monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply-shaped convex curve. We also obtain a variant of the Gallai-Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.

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“…´ Ä µ as in Section 2.1. We draw in the plane using the same drawing rule described in Section 2.1, 5 We partition into two subsets ½ ¾ of size at most Ò ¾ each so…”

Section: ¾ 4 Improving the Tamaki-tokuyama Boundmentioning

confidence: 99%

Lenses in arrangements of pseudo-circles and their applications

Agarwal

Nevo

Pach

et al. 2004

J. ACM

161

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“…Those include a version for set systems with bounded VC-dimension [Mat04], colorful and fractional versions [BFM + 14] and a generalization to a topological (p, q)-theorem for finite families of sets which are so-called good cover, i.e., the intersection of every sub-family is either empty or contractible [AKMM01].…”

Section: Introductionmentioning

confidence: 99%

On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Keller¹,

Smorodinsky²,

Tardos³

2017

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

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Let HD d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in R d which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved thatThis paper has two parts. In the first part we present several improved bounds on HD d (p, q). In particular, we obtain the first near tight estimate of HD d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem.In the second part we prove a (p, 2)-theorem for families in R 2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

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“…Indeed, this makes the task of finding a small net much easier, and unlike the case of strong nets it is known that there is a function f ( , m) depending only on and m so that for every finite set X (of any size) in R m , there is a weak -net Y for X with respect to the set of all convex sets in R m , where |Y | ≤ f ( , m). This was first proved in [2], see also [9] and [24] for improved bounds, and [4] and its references for several extensions. The corresponding assertion for strong nets is easily seen to be false (and indeed the VC-dimension of the family of all convex sets is infinite, in every fixed dimension m ≥ 2).…”

Section: Weak -Netsmentioning

confidence: 88%

“…As shown in [20] there are known constructions in which for fixed d the size of the smallest possible -net for a given set cannot be linear in 1/ . In fact, the O( d log(1/ )) bound may be tight already for dimension d = 2, as shown in [20] (see also [4] for another construction). Despite the existence of these constructions, there is no known natural geometric example demonstrating this phenomenon.…”

Section: Introductionmentioning

confidence: 83%