A Simple Proof of the Shallow Packing Lemma

International audienceWe show that the shallow packing lemma follows from a simple modification of the standard proof, due to Haussler and simplified by Chazelle, of the packing lemma

Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications

Given a set system (X, S), constructing a matching of X with low crossing number is a key tool in combinatorics and algorithms. In this paper we present a new sampling-based algorithm which is applicable to finite set systems. Let n = |X|, m = | S| and assume that X has a perfect matching M such that any set in ? crosses at most ? = ?(n^?) edges of M. In the case ? = 1- 1/d, our algorithm computes a perfect matching of X with expected crossing number at most 10 ?, in expected time O? (n^{2+(2/d)} + mn^(2/d)). As an immediate consequence, we get improved bounds for constructing low-crossing matchings for a slew of both abstract and geometric problems, including many basic geometric set systems (e.g., balls in ?^d). This further implies improved algorithms for many well-studied problems such as construction of ?-approximations. Our work is related to two earlier themes: the work of Varadarajan (STOC \u2710) / Chan et al. (SODA \u2712) that avoids spatial partitionings for constructing ?-nets, and of Chan (DCG \u2712) that gives an optimal algorithm for matchings with respect to hyperplanes in ?^d. Another major advantage of our method is its simplicity. An implementation of a variant of our algorithm in C++ is available on Github; it is approximately 200 lines of basic code without any non-trivial data-structure. Since the start of the study of matchings with low-crossing numbers with respect to half-spaces in the 1980s, this is the first implementation made possible for dimensions larger than 2

A Theorem of Barany Revisited and Extended

International audienceThe colorful Caratheodory theorem states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Caratheodory theorem: given d/2+1 sets of points in $R^d, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a (d/2+1)-dimensional rainbow simplex intersecting C

An Optimal Generalization of the Colorful Carathéodory Theorem

International audienceThe Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C

Optimality of Geometric Local Search

International audienceUp until a decade ago, the algorithmic status of several basic NP-complete problems in geometric combinatorial optimisation was unresolved. This included the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. These past nine years have seen the resolution of all these problems—interestingly, with the same algorithm: local search. In fact, it was shown that for many of these problems, local search with radius λ gives a (1 + O(λ − 1 2))-approximation with running time n O(λ). Setting λ = Θ(epsilon^{−2}) yields a PTAS with a running time of n^O(epsilon^{−2}). On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n)·f () for any arbitrary f. Thus the main question left open in previous work is in improving the exponent of n to o(epsilon^{−2}). We show that in fact the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility results. Our construction extends to other graph families with small separators. Acknowledgements We thank the referees for several helpful comments

Optimal Approximations Made Easy

The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of this paper is to give a modular, self-contained, intuitive proof of this result for finite set systems. The only ingredient we assume is the standard Chernoff's concentration bound. This makes the proof accessible to a wider audience, readers not familiar with techniques from statistical learning theory, and makes it possible to be covered in a single self-contained lecture in a geometry, algorithms or combinatorics course.Comment: 7 page