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

The ?-t-Net Problem

We study a natural generalization of the classical ?-net problem (Haussler - Welzl 1987), which we call the ?-t-net problem: Given a hypergraph on n vertices and parameters t and ? ? t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ? n contains a set in S. When t=1, this corresponds to the ?-net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an ?-t-net of size O((1+log t)d/? log 1/?). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/?)-sized ?-t-nets. We also present an explicit construction of ?-t-nets (including ?-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ?-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest

Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In this paper we present several new results and applications related to packings: * an optimal lower bound for shallow packings, * improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry, * we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and * simplifying and generalizing one of the main technical tools in [Fox et al.J. of the EMS, to appear]

Sur les algorithmes d'approximation combinatoires en géométrie

The analysis of approximation techniques is a key topic in computational geometry, both for practical and theoretical reasons. In this thesis we discuss sampling tools for geometric structures and geometric approximation algorithms in combinatorial optimization. Part I focuses on the combinatorics of geometric set systems. We start by discussing packing problems in set systems, including extensions of a lemma of Haussler, mainly the so-called shallow packing lemma. For said lemma we also give an optimal lower bound that had been conjectured but not established in previous work on the topic. Then we use this lemma, together with the recently introduced polynomial partitioning technique, to study a combinatorial analogue of the Macbeath regions from convex geometry: Mnets, for which we unify previous existence results and upper bounds, and also give some lower bounds. We highlight their connection with epsilon-nets, staples of computational and combinatorial geometry, for example by observing that the unweighted epsilon-net bound of Chan et al. (SODA 2012) or Varadarajan (STOC 2010) follows directly from our results on Mnets. Part II deals with local-search techniques applied to geometric restrictions of classical combinatorial optimization problems. Over the last ten years such techniques have produced the first polynomial-time approximation schemes for various problems, such as that of computing a minimum-sized hitting set for a collection of input disks from a set of input points. In fact, it was shown that for many of these problems, local search with radius Θ(1/epsilon²) gives a (1 + epsilon)-approximation with running time n^{O(1/epsilon²)}. However the question of whether the exponent of n could be decreased to o(1/epsilon²) was left open. We answer it in the negative: the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility resultsL'analyse des techniques d'approximation est centrale en géométrie algorithmique, pour des raisons pratiques comme théoriques. Dans cette thèse nous traitons de l'échantillonnage des structures géométriques et des algorithmes d'approximation géométriques en optimisation combinatoire. La première partie est consacrée à la combinatoire des hypergraphes. Nous débutons par les problèmes de packing, dont des extensions d'un lemme de Haussler, particulièrement le lemme dit de Shallow packing, pour lequel nous donnons aussi un minorant optimal, conjecturé mais pas établi dans les travaux antérieurs. Puis nous appliquons ledit lemme, avec la méthode de partition polynomiale récemment introduite, à l'étude d'un analogue combinatoire des régions de Macbeath de la géométrie convexe : les M-réseaux, pour lesquels nous unifions les résultats d'existence et majorations existants, et donnons aussi quelques minorants. Nous illustrons leur relation aux epsilon-réseaux, structures incontournables en géométrie combinatoire et algorithmique, notamment en observant que les majorants de Chan et al. (SODA 2012) ou Varadarajan (STOC 2010) pour les epsilon-réseaux (uniformes) découlent directement de nos résultats sur les M-réseaux. La deuxième partie traite des techniques de recherche locale appliquées aux restrictions géométriques de problèmes classiques d'optimisation combinatoire. En dix ans, ces techniques ont produit les premiers schémas d'approximation en temps polynomial pour divers problèmes tels que celui de calculer un plus petit ensemble intersectant pour un ensemble de disques donnés en entrée parmi un ensemble de points donnés en entrée. En fait, il a été montré que pour de nombreux tels problèmes, la recherche locale de rayon Θ (1/epsilon²) donne une (1 + epsilon)-approximation en temps n^{O(1/epsilon²)}. Savoir si l'exposant de n pouvait être ramené à o (1/epsilon²) demeurait une question ouverte. Nous répondons par la négative : la garantie d'approximation de la recherche locale n'est améliorable pour aucun desdits problème

Shallow packings, semialgebraic set systems, Macbeath regions, and polynomial partitioning

International audienceGiven a set system (X,R) such that every pair of sets in R have large symmetric difference, the Shallow Packing Lemma gives an upper bound on |R| as a function of the shallow-cell complexity of R. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system (X,R) with shallow-cell complexity φ(⋅,⋅) and a parameter ϵ>0, there exists a collection, called an ϵ-Mnet, consisting of $O(1ϵφ(O(1ϵ),O(1)))$ subsets of X, each of size $Ω(ϵ|X|)$, such that any $R∈R$ with $|R|≥ϵ|X|$ contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal ϵ-net bound follows