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13
Quadratic forms on graphs
 Invent. Math
, 2005
"... We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R, ..."
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Cited by 32 (10 self)
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We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R,
Transversal Numbers for Hypergraphs Arising in Geometry
 Adv. Appl. Math
, 2001
"... Introduction Helly's theorem asserts that if F is a finite family of convex sets in R d in which every d + 1 or fewer sets have a point in common then T F 6= ;. Our starting point, the (p; q) theorem, is a deep extension of Helly's theorem. It was conjectured by Hadwiger and Debrunner and prove ..."
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Cited by 16 (3 self)
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Introduction Helly's theorem asserts that if F is a finite family of convex sets in R d in which every d + 1 or fewer sets have a point in common then T F 6= ;. Our starting point, the (p; q) theorem, is a deep extension of Helly's theorem. It was conjectured by Hadwiger and Debrunner and proved by Alon and Kleitman [3]. Let p q 2 be integers. A family F of convex sets in R d is said to have the (p; q) property if among every p sets of F , some q have a point in common. Theorem 1 ((p; q) theorem, Alon & Kleitmen) For every p q d+ 1 there exists a number C =
Fast stabbing of boxes in high dimensions
, 2000
"... We present in this paper a simple yet efficient algorithm for stabbing a set S of n axisparallel boxes in ddimensional space with c(S) points in outputsensitive time O(dn log c(S)) and linear space. Let c ∗ (S) and b ∗ (S) be, respectively, the minimum number of points required to stab S and the ..."
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Cited by 16 (3 self)
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We present in this paper a simple yet efficient algorithm for stabbing a set S of n axisparallel boxes in ddimensional space with c(S) points in outputsensitive time O(dn log c(S)) and linear space. Let c ∗ (S) and b ∗ (S) be, respectively, the minimum number of points required to stab S and the maximum number of pairwise disjoint boxes of S. We prove that b ∗ (S)6c ∗ (S)6c(S)6b ∗ (S)(1+log2 b ∗ (S)) d−1. Since nding a minimal set of c ∗ (S) points is NPcomplete as soon as d¿1, we obtain a fast precisionsensitive heuristic for stabbing S whose quality does not depend on the input size. In the case of congruent or constrained isothetic boxes, our algorithm reports, respectively, c(S)62 d−1 b ∗ (S) and c(S)=Od(b ∗ (S)) stabbing points. Moreover, we show that the bounds we get on c(S) are asymptotically tight and corroborate our results with some experiments. We also describe an optimal outputsensitive algorithm for nding a minimalsize optimal stabbing pointset of intervals. Finally, we conclude with
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
On Point Covers Of Multiple Intervals And Axisparallel Rectangles
, 1995
"... In certain families of hypergraphs the transversal number is bounded by some function of the packing number. In this paper we study hypergraphs related to multiple intervals and axisparallel rectangles, respectively. Essential improvements of former established upper bounds are presented here. We e ..."
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Cited by 4 (0 self)
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In certain families of hypergraphs the transversal number is bounded by some function of the packing number. In this paper we study hypergraphs related to multiple intervals and axisparallel rectangles, respectively. Essential improvements of former established upper bounds are presented here. We explore the close connection between the two problems at issue.
VapnikChervonenkis dimension and (pseudo)hyperplane arrangements
, 1997
"... An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let ..."
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Cited by 4 (1 self)
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An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of R is maximum for the given VCdimension. In general, such range spaces are called maximum, and we show that the number of ranges R ∈ R for which also X −R ∈ R, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and ‘small ’ subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniform oriented matroids: a range space (X, R) naturally corresponds to a uniform oriented matroid of rank X  − d if and only if its VCdimension is d, R ∈ R implies X − R ∈ R and R  is maximum under these conditions.
Dominating sets in kmajority tournaments
 J. Combin. Theory Ser. B
, 2006
"... A kmajority tournament T on a finite vertex set V is defined by a set of 2k − 1 linear orderings of V, with u → v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of “nontransitive dice”, we let F (k) be the maximum over all kmajority tournaments T of ..."
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Cited by 4 (0 self)
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A kmajority tournament T on a finite vertex set V is defined by a set of 2k − 1 linear orderings of V, with u → v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of “nontransitive dice”, we let F (k) be the maximum over all kmajority tournaments T of the size of a minimum dominating set of T. We show that F (k) exists for all k> 0, that F (2) = 3 and that in general C1k / log k ≤ F (k) ≤ C2k log k for suitable positive constants C1 and C2. 1
On point covers of coriented polygons
, 2001
"... Let S be any family of n coriented polygons of the twodimensional Euclidean plane E 2, i.e., bounded intersection of halfplanes whose normal directions of edges belong to a fixed collection of c distinct directions. Let (S) denote the packing number of S, that is the maximum number of pairwise dis ..."
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Cited by 3 (1 self)
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Let S be any family of n coriented polygons of the twodimensional Euclidean plane E 2, i.e., bounded intersection of halfplanes whose normal directions of edges belong to a fixed collection of c distinct directions. Let (S) denote the packing number of S, that is the maximum number of pairwise disjoint objects of S. Let (S) be the transversal number of S, that is the minimum number of points required so that each object contains at least one of those points. We prove that (S)6G(2;c) (S) log c−1 2 ( (S)+1), where G(2;c) is the Gallai number of pairwise intersecting coriented polygons. Our bound collapses to (S)=O(G(2;c) (S)) if objects are more or less of the same size. We describe a t(n; c)+O(nc log (S))time algorithm with linear storage that computes such a 0transversal, where t(n; c) is the time required to pierce pairwise intersecting coriented polygons. We provide lineartime algorithms t(n; c) = (nc) forfat coriented polytopes, translates or homothets of E d proving that G(2;c)=O ( ) d, G(2;c)6d d
Geometric Set Systems
, 1998
"... Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, e ..."
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Cited by 2 (0 self)
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Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the VapnikChervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...
A Remark on Transversal Numbers
, 1997
"... Introduction In his classical monograph published in 1935, Denes Konig [K] included one of Paul Erd}os's rst remarkable results: an in nite version of the Menger theorem. This result (as well as the Konig{Hall theorem for bipartite graphs, and many related results covered in the book) can be ref ..."
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Introduction In his classical monograph published in 1935, Denes Konig [K] included one of Paul Erd}os's rst remarkable results: an in nite version of the Menger theorem. This result (as well as the Konig{Hall theorem for bipartite graphs, and many related results covered in the book) can be reformulated as a statement about transversals of certain hypergraphs. Let H be a hypergraph with vertex set V (H) and edge set E(H). A subset T V (H) is called a transversal of H if it meets every edge E 2 E(H). The transversal number (H) is de ned as the minimum cardinality of a transversal of H. Clearly, (H) (H), where (H) denotes the maximum number of pairwise disjoint edges of H. In the above mentioned examples, (H) = (H) holds for the corresponding hypergraphs. However, in general it is impossible to bound from above by any function of , without putting some restriction on the structure of H. One of Erd}os's closest friends and collaborators, Tibor Gallai (who is also quo