On a Problem of Danzer

Transcription

1 On a Problem of Danzer Nabil H. Mustafa Université Paris-Est, Laboratoire Informatiue Gaspar-Monge, Euipe A3SI, ESIEE Paris mustafan@esiee.fr Saurabh Ray Department of Computer Science, NYU Abu Dhabi, Unite Arab Emirates saurabh.ray@nyu.eu Abstract Let C be a boune convex object in R, an P a set of n points lying outsie C. Further let c p, c be two integers with c c p n, such that every cp + points of P contains a subset of size c + whose convex-hull is isjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex-hull is isjoint from C. Our proof is constructive an implies that such a partition can be compute in polynomial time. In particular, our general theorem implies polynomial bouns for Hawiger-Debrunner p, numbers for balls in R. For example, it follows from our theorem that when p > + β for β > 0, then any set of balls satisfying the HDp, property can be hit by O p + β log p points. This is the first improvement over a nearly 60-year ol exponential boun of roughly O. Our results also complement the results obtaine in a recent work of Keller et al. where, apart from improvements to the boun on HDp, for convex sets in R for various ranges of p an, a polynomial boun is obtaine for regions with low union complexity in the plane. 0 ACM Subject Classification Theory of computation Computational geometry Keywors an phrases Convex polytopes, Hawiger-Debrunner numbers, Epsilon-nets, Balls Digital Object Ientifier 0.430/LIPIcs.ESA Introuction Given a finite set C of geometric objects in R, we say that C satisfies the HDp, property if for any set C C of size p, there exists a point in R common to at least objects of C. The goal then is to show that there exists a small set Q of points in R such that each object of C contains some point of Q; such a Q is calle a hitting set for C. These bouns for a set C of convex sets in R have been stuie since the 950s see the surveys [7, 8, 5], an it was only in 99 that Alon an Kleitman [], in a breakthrough result, gave an upper-boun that is inepenent of C. Unfortunately it epens exponentially on p, an. For the case where C consists of arbitrary convex objects, the current best bouns remain exponential in p, an. The work of Nabil H. Mustafa in this paper has been supporte by the grant ANR SAGA JCJC-4- CE Nabil H. Mustafa an Saurabh Ray; license uner Creative Commons License CC-BY 6th Annual European Symposium on Algorithms ESA 08. Eitors: Yossi Azar, Hannah Bast, an Grzegorz Herman; Article No. 64; pp. 64: 64:8 Leibniz International Proceeings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 64: On a Problem of Danzer Theorem A [, 9]. Let C be a finite set of convex objects in R satisfying the HDp, property, where p, are two integers with p +. Then there exists a hitting set for C of size O p p + O p +, log c 3 log p p, log c 3 log p, for log p for p +ɛ, p p, ɛ. where c is an absolute constant inepenent of C, p, an, an p, ɛ is a function epening only on an ɛ. Consier the basic case where C is a set of balls in R satisfying the HDp, property. Theorem A implies ignoring logarithmic factors an for general values of p an the existence of a hitting set of size no better than O p. Furthermore, it reuires + a necessary conition for arbitrary convex objects but not for balls. Almost 60 years ago, Danzer [4, 5] consiere the HDp, problem for balls. The best boun that we are aware of, erive from the survey of Eckhoff [7] by combining ineualities 4., 4.4 an 4.5, is state below. It is better than the one from Theorem A uantitatively, but also in that it gives a boun reuiring only that. Further, for a very specific case namely when p = an is Olog it succees in giving polynomial bouns. Theorem B [7]. Let B be a finite set of balls in R. If B satisfies the HDp, property for some p, then there exists a hitting set for B of size at most 3π p 3 3 g g + where gx = log x + log log x +. Ignoring logarithmic terms, the above boun is of the form Θ p If p the first term ominates, otherwise the secon term ominates. Turning towars the lower-boun for the case where C is a set of unit balls in R, Bourgain an Linenstrauss [] prove a lower-boun of.0645 when p = = in R, i.e., one nees at least.0645 points to hit all pairwise intersecting unit balls in R. Our Result We consier a more general set up for the HDp, problem, as follows. Let C be a convex object in R, an P a set of n points lying outsie C. For each p P, let H p be the set of hyperplanes separating p from C. Let C p be the set of points in R ual to the hyperplanes in H p see [, Chapter 5.], an let S = {C p : p P }. Our goal is to stuy the HDp, property for S namely, that out of every p objects of S, there exists a point in R common to at least of them. This is euivalent to the property of C an P that out of every p-size set P P, there exists a hyperplane separating C from a -size subset P P or euivalently, convp is isjoint from C. Our main theorem is the following. For a simpler expression, let c, c p be two positive integers such that p = c p + an = +. There are easy examples, e.g. when the convex objects are hyperplanes in R.

3 N. H. Mustafa an S.Ray 64:3 Theorem. Let C be a boune convex object in R an P a set of n points lying outsie C. Further let c p, c be two integers, with c c p n, such that for every cp + points of P, there exists a subset of size c + whose convex-hull is isjoint from C. Then the points of P can be partitione into λ c p, c = K K c / + c / + / + c p log / + c p sets, each of whose convex-hull is isjoint from C. Here K, K are absolute constants inepenent of n,, c p an c. Furthermore, such a partition can be compute in polynomial time. The proof, presente in Section, is a combination of three ingreients: the Alon-Kleitman techniue [], bouns on inepenent sets in hypergraphs [9] an bouns on k-sets for half-spaces [3]. It is an extension of the proof in [4] which stuie Carathéoory s theorem in this setting. Remark. The restriction that + is necessary as can be seen when P form the vertices of a cyclic polytope in R an C is a slightly shrunk copy of convp. Remark. Note that when c β for any absolute constant β > 0, the above boun is polynomial in the imension it is upper-boune by O p + β log p. Remark. It was shown in [3] that C p is a convex object in R an thus the bouns of Theorem A apply. As before, Theorem substantially improves upon this, as the bouns following from Theorem A are exponential in an furthermore, reuire +. As an immeiate corollary of Theorem, we obtain the first improvements to the ol boun on the p, problem for balls in R. The boun in Theorem B is exponential in except in special cases where p = an is 3 Olog. On the other han, our result gives polynomial bouns as long as β for any constant β >. Corollary Hawiger-Debrunner p, boun for balls in R. Let B be collection of balls in R such that for every subset of c p + + balls in B, some + + have a common. Then there intersection, where c p an c are integers such that c c p n + exists a set X of λ + c p, c points that form a hitting set for the balls in B. Here λ +, is the function efine in the statement of Theorem. Proof. Observe that one can stereographically lift balls in R to caps of a sphere S in R +, where a cap of a sphere is a portion of the sphere containe in a half-space that oesn t contain the center of the sphere. Thus we will prove a slightly more general result where B consists of caps of a -imensional sphere S embee in R +. For a point x S, let h x enote the hyperplane tangent to S at x. For any point y lying outsie S, efine the separating set of y to be S y = {z S : h z separates y an S}. Geometrically, S y is the set of points of S visible from y, an form a cap of S. Furthermore, for any cap K of S, there is a uniue point w such that K = S w. We enote this point w by apexk. 3 Recall that Theorem B assumes p. E S A 0 8

4 64:4 On a Problem of Danzer Given the set of caps B on S, consier the point set apex B = {apexb: B B}. Observe that for a point x S an a cap B B, x B if an only if x S apexb. As B satisfies the p, property namely that for every p-size subset B of B, there exists a point x S lying in some elements of B we have that for every p-size subset A of apexb, there exists a point x S lying in the separating set of some points of A. In other wors, h x separates these points from S. Applying Theorem with C = S an P = apex B in imension +, we conclue that P can be partitione into a family Ξ of λ + c p, c sets, each of whose convex hull is isjoint from S. Consier a set P Ξ. Since the convex hull of P is isjoint from S, we can fin a hyperplane h x tangent to S at x such that h x separates P from S. This implies that all the caps in B corresponing to the points in P contain the point x. Thus for each set of Ξ we obtain a point which is containe in all the caps corresponing to the points in that set. These X = λ + c p, c points form the reuire set X. Our results complement the recent results of Keller, Smoroinsky an Taros [9, 0] who obtain polynomial bouns for regions of low union complexity in the plane. Proof of Theorem Given a set P of points in R an an integer k, a set P P is calle a k-set of P if P = k an if there exists a half-space h in R such that P = P h. A set P P is calle a k-set if P is a l-set for some l k. The next well-known theorem gives an upper-boun on the number of k-sets in a point set see [7]. Theorem 3 Clarkson-Shor [3]. For any integer k +, the number of k-sets of any set of n points in R is at most / K n κ n, k = k + / / κ k n /, / / where κ k = K / / + k / / an K e is an absolute constant inepenent of n, an k. Definition 4 Depth. Given a set P of n points in R an any set Q P, efine the epth of Q with respect to P, enote epth P Q, to be the minimum number of points of P containe in any half-space containing Q. For two parameters l k, let τ n, k, l enote the maximum number of subsets of size k an epth at most l with respect to P in any set P of n points in R : τ n, k, l = max {Q P : Q = k an epth P Q l}. P R P =n The following statement is easily implie by an application of the Clarkson-Shor techniue [3] e.g., see [6]. Theorem 5. For parameters l k +, τ n, k, l e κ n, k l k /, where the function κ, is as efine in Euation.

5 N. H. Mustafa an S.Ray 64:5 Proof. Let P be any set of n points in R. Let t be the number of sets of P of size k an epth at most l. Pick each element of P inepenently with probability ρ = l to get a ranom sample R. The expecte number of k-sets in R satisfies ρ k ρ l k t E [ number of k-sets in R ] K [ ] R E / k + K n = / ρ k + = κ n, k ρ = t κ n, k ρ ρ k ρ l k e κ n, k l k /, as l l k e for any l k. Lemma 6. Let C be a boune convex object in R, an P a set of n points lying outsie C. Let p + be parameters such that for every subset Q P of size p, there exists a set Q Q of size such that Q can be separate from C by a hyperplane. Then there exists a hyperplane separating at least p e κ / fraction of the points of P from C. Proof. From [6, 9], it follows that the number of istinct -tuples of P that can be separate from C by a hyperplane is, assuming that n p, at least n p + n + n p n p. Let l be the maximum epth Definition 4 of any of these separable -tuples. The number of such tuples is therefore at most τ n,, l. Thus by Theorem 5 we must have n p τ n,, l e κ n, l /. Re-arranging the terms an from ineuality, we get l n p e κ n, / n p e κ n / = n p e κ /. Thus one of the separable -tuples, say P P, must have epth at least l; in other wors, the hyperplane separating P from C must contain at least l points of P. This is the reuire hyperplane. We now prove a weighte version of the above statement. E S A 0 8

6 64:6 On a Problem of Danzer Corollary 7. Let C be a boune convex object in R, an P a weighte set of n points lying outsie C, where the weight of each point p P is a non-negative rational number. Let p + be parameters such that for every subset Q P of size p, there exists a set Q Q of size such that Q can be separate from C by a hyperplane. Then there exists a hyperplane separating a set of points whose weight is at least α p, = e κ p / fraction of the total weight of the points in P. Proof. By appropriately scaling all the rational weights, we may assume that each weight is a non-negative integer an we replace a point with weight m by m unweighte copies of the point. Let P be the new set of points. Observe that any set S of p points in P either contains copies of some point in P or it contains p istinct points from P. In either case, there is hyperplane separating points of S from C. Thus, we can apply Lemma 6 to the point set P with the parameter p in the lemma replace by p. The result follows. Finally we return to the proof of the main theorem. Theorem. Let C be a boune convex object in R an P a set of n points lying outsie C. Further let c p, c be two integers, with c c p n, such that for every cp + points of P, there exists a subset of size c + whose convex-hull is isjoint from C. Then the points of P can be partitione into λ c p, c = K K c / + c / + / + c p log / + c p sets, each of whose convex-hull is isjoint from C. Here K, K are absolute constants inepenent of n,, c p an c. Furthermore, such a partition can be compute in polynomial time. Proof. Let p = c p + / an = c + /. Let H be the set of all hyperplanes separating a istinct subset of points of P from C. As the number of subsets of P is finite, one can assume that H is also finite. Consier the following linear program on H variables {u h 0: h H}: min h H u h, such that r P : h H h separates r from C The LP-ual to the above program, on P variables {w r 0: r P }, is: max p P w p, such that h H: r P h separates r from C u h. w r. 3 Consier an optimal solution w of the ual linear program an interpret wr as the weight of each r P. Since the weights are rational, by Corollary 7, there exists a hyperplane h H separating a subset of P of combine weight at least ɛ = α p, fraction of the total weight of all the points. Since the total weight of the points in any half-space is constraine to be at most by the linear program, the total weight of all the points of P must be at most ɛ. In other wors, the optimal value of linear program 3 is at most ɛ. Since the optimal values of both linear programs are eual ue to strong uality, the optimal value of linear program is also at most ɛ.

7 N. H. Mustafa an S.Ray 64:7 Let u be the optimal solution of linear program. If we interpret u h as the weight of the hyperplane h, the constraints of the program imply that each point is separate by a set of hyperplanes in H whose combine weight is at least out of a total weight of at most ɛ in other wors, at least ɛ-th fraction of the total weight of H. By associating with each hyperplane the half-space boune by it an not containing C, an using the ɛ-net theorem for half-spaces in R see [], there exists a set of O ɛ log ɛ hyperplanes which together separate all points of P from C. Recalling that ɛ = α p, = e κ p / = e κ p. an that κ = K / / + / /, we get 4K ɛ = e / / + / p / 4K + / c + / p 4K + / c + / + / p + / = O K / + / = O K = O K = O K = O K / + / + / + = O K + c / e / using e K an = c + / / + c + / c p + / + c / + + c p / / + c / e / / + c p + c + c / + p / / / + c / + c p / + c / + c p / + / +. / + / / + / + The Big-Oh notation here oes not hie epenencies on namely we o not treat as a constant. From the above it follows that log ɛ = O c / + c log / + c p. Thus, ɛ log ɛ is K O / + c / + c p / + c / + c log / + c p which simplifies to O K / + c / + / + c p log / + c p. c Since linear programs can be solve in polynomial time an epsilon nets can be compute in polynomial time, the partition of P into the above number of sets can be achieve in polynomial time. The theorem follows. E S A 0 8

8 64:8 On a Problem of Danzer References N. Alon an D. Kleitman. Piercing convex sets an the Hawiger Debrunner p, -problem. Av. Math., 96:03, 99. J. Bourgain an J. Linenstrauss. On covering a set in R n by balls of the same iameter. In Geometric Aspects of Functional Analysis, pages Springer Berlin Heielberg, K. Clarkson an P. Shor. Applications of ranom sampling in computational geometry, II. Discrete & Computational Geometry, 45:387 4, L. Danzer. Uber zwei Lagerungsprobleme; Abwanlungen einer Vermutung von T. Gallai. PhD thesis, Techn. Hochschule Munchen, L. Danzer. Uber urchschnittseigenschaften n-imensionaler kugelfamilien. J. Reine Angew. Math., 08:8 03, D. e Caen. Extension of a theorem of Moon an Moser on complete subgraphs. Ars Combin., 6:5 0, J. Eckhoff. A survey of the Hawiger-Debrunner p, -problem. In Discrete an Computational Geometry: The Gooman-Pollack Festschrift, pages Springer, A. Holmsen an R. Wenger. Helly-type theorems an geometric transversals. In J. E. Gooman, J. O Rourke, an C. D. Tóth, eitors, Hanbook of Discrete an Computational Geometry, pages 9 3. CRC Press LLC, C. Keller, S. Smoroinsky, an G. Taros. Improve bouns on the Hawiger-Debrunner numbers. Israel J. of Math., to appear, C. Keller, S. Smoroinsky, an G. Taros. On max-cliue for intersection graphs of sets an the hawiger-ebrunner numbers. In Proceeings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms SODA, pages 54 63, 07. A. Kupavskii, N. H. Mustafa, an J. Pach. Near-optimal lower bouns for ɛ-nets for halfspaces an low complexity set systems. In Martin Loebl, Jaroslav Nešetřil, an Robin Thomas, eitors, A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek, pages Springer International Publishing, 07. J. Matoušek. Lectures in Discrete Geometry. Springer-Verlag, New York, NY, N. H. Mustafa an S. Ray. Weak ɛ-nets have a basis of size O/ɛ log /ɛ. Comp. Geom: Theory an Appl., 40:84 9, N. H. Mustafa an S. Ray. An optimal generalization of the colorful Carathéoory theorem. Discrete Mathematics, 3394: , N. H. Mustafa an K. Varaarajan. Epsilon-approximations an epsilon-nets. In J. E. Gooman, J. O Rourke, an C. D. Tóth, eitors, Hanbook of Discrete an Computational Geometry, pages CRC Press LLC, S. Smoroinsky, M. Sulovský, an U. Wagner. On center regions an balls containing many points. In Proceeings of the 4th annual International Conference on Computing an Combinatorics, COCOON 08, pages , U. Wagner. k-sets an k-facets. In J.E. Gooman, J. Pach, an R. Pollack, eitors, Surveys on Discrete an Computational Geometry: Twenty Years Later, Contemporary Mathematics, pages American Mathematical Society, 008.