Hardness and Approximation of Minimum Convex Partition

We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an O(log OPT)-approximation running in O(n^8)-time, where OPT denotes the minimum number of convex faces needed, to an O(sqrt(n) log n)-approximation algorithm running in O(n^2)-time. We say that a point set is k-directed if the (straight) lines containing at least three points have up to k directions. We present an O(k)-approximation algorithm running in n^O(k)-time. Those hardness and approximation results also holds for the Minimum Convex Tiling problem, defined similarly but allowing the use of Steiner points. The approximation results are obtained by relating the problem to the Covering Points with Non-Crossing Segments problem. We show that this problem is NP-hard, and present an FPT algorithm. This allows us to obtain a constant-approximation FPT algorithm for the Minimum Convex Partition Problem where the parameter is the number of faces.Comment: Addition of algorithms with better approximation ratio

Maximum Clique in Generalisations of Disk Graphs and Plane Geometric Graphs on Degenerate Point Sets

This thesis deals with graphs having geometric representations. On one hand we consider graphs whose vertices can be mapped to geometric objets in an Euclidean space (for instance disks in the plane), such that two vertices are adjacent if and only if the corresponding objects intersect. Those are called "intersection graphs'', and if all objects are constrained to be, e.g. disks, then they are referred to as "disk graphs''. On the other hand we study graphs that can be represented in the plane such that vertices are mapped to points and edges to straight-line segments, such that no two edges cross. Those are called "plane geometric graphs''. In this thesis, we also have additional conditions. We consider for instance convex partitions, for which the union of the faces is equal to the convex hull of the points and each bounded face is convex. As a special case, we also study triangulations, for which we additionally require that every bounded face be a triangle. For intersection graphs, we study the problem of finding a maximum clique in disk-like intersection graphs. In 1990, a seminal paper by Clark, Colbourn and Johnson showed that maximum clique can be solved in polynomial time in unit disk graphs. However, the complexity of maximum clique in disk graphs is still unknown. Recently, Bonamy et al.\ showed the existence of an EPTAS for maximum clique in disk graphs. This leads to the following questions: Are there superclasses of unit disk graphs in which maximum clique can be solved in polynomial time? Are there superclasses of disk graphs for which there is an EPTAS? Are there related classes for which we can show NP-hardness? Concerning the first question, we show that maximum clique can be solved in polynomial time in intersection graphs of translates of a fixed bounded convex set. Furthermore, we define a superclass C of both unit disk graphs and interval graphs, where C is defined as the intersection graph class of some specified sets, for which there exists a polynomial time algorithm. For the second question, we prove the existence of an EPTAS for homothets of a fixed bounded and centrally symmetric convex set. We also give partial results toward showing the existence of an EPTAS for intersection graphs of convex pseudo-disks. Finally, for the third question, we show that maximum clique is NP-hard, and even APX-hard, in intersection graphs of unit disks and axis-parallel rectangles. Concerning triangulations and convex partitions, we study two problems that were previously only considered under the assumption that no three of the n input points are on a line. For triangulations, we extend a result by Wagner and Welzl and show that the bistellar flip graph is (n-3)-connected. For convex partitions, we provide the first approximation algorithms for computing convex partitions with as few faces as possibles when three points or more may lie on a line. In particular, we give an O(log(OPT))-approximation algorithm running in time O(n^8), where OPT denotes the size of a minimum solution. We additionally provide an O(sqrt(n)log(n))-approximation algorithm running in time O(n^2). We also show that minimising the number of faces is NP-hard

Computing a maximum clique in geometric superclasses of disk graphs

In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.ISSN:1382-6905ISSN:1573-288

Conception, étude et réduction de l'hydrophilie d'emballages antimicrobiens à base de papier et de chitosane

Les emballages alimentaires antimicrobiens élaborés à partir de ressources renouvelables constituent une réponse aux problèmes de santé publique causés par les intoxications alimentaires, et aux problèmes de gestion des déchets d'emballages en matières plastiques synthétiques. L'élaboration de tels emballages a donc été envisagée ici sous la forme d'un système de papiers enduits de chitosane, un biopolymère à l'activité antimicrobienne démontrée. Cette étude s'est attachée à réduire le Coefficient de Transfert à la Vapeur d'Eau de tels matériaux, tout en conservant leur activité biologique. Deux approches ont pour cela été mises en place : Une approche "matériaux" consistant à enduire un mélange simple de chitosane et d'acide gras sur les papiers, Une approche chimique consistant à modifier chimiquement le chitosane tout en préservant son activité antimicrobienne. L'étude de l'influence de ces deux approches sur la structure des matériaux, sur la bioactivité et sur les propriétés classiques des emballages alimentaires a ensuite été réalisée (sensibilité à l'eau liquide, propriétés mécaniques, coefficient de transfert à l'oxygène). L'enduction du mélange simple chitosane-acide gras et l'utilisation de chitosane O-acylé n'ont pas permis de réduire le CTVE des matériaux, mais ceux-ci ont montré une activité efficace contre Listeria monocytogenes et Salmonella typhimurium ainsi qu'une résistance à l'eau liquide et des propriétés mécaniques intéressantes. Par ailleurs, pour la première fois, un dérivé du chitosane potentiellement bioactif a été oxydé en position C6 par le TEMPO. Enfin, une étude fondamentale des interactions eau-matériaux a été réalisée par RMN-basse résolution, conduisant à des corrélations avec les propriétés macroscopiques des emballages.BORDEAUX1-BU Sciences-Talence (335222101) / SudocSudocFranceF

Well-Separation and Hyperplane Transversals in High Dimensions

A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ?^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an ?((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete. Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ?^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ? d+1 finite point sets in ?^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d

Nearest-Neighbor Decompositions of Drawings

P_c. We show that it is NP-complete to decide whether ? can be drawn as the union of c ? 3 nearest-neighbor graphs, even when no two segments cross. We show that for c = 2, it is NP-complete in the general setting and polynomial-time solvable when no two segments cross. We show the existence of an O(log n)-approximation algorithm running in subexponential time for partitioning ? into a minimum number of nearest-neighbor graphs. As a main tool in our analysis, we establish the notion of the conflict graph for a drawing ?. The vertices of the conflict graph are the connected components of ?, with the assumption that each connected component is the nearest neighbor graph of its vertices, and there is an edge between two components U and V if and only if the nearest neighbor graph of U ? V contains an edge between a vertex in U and a vertex in V. We show that string graphs are conflict graphs of certain planar drawings. For planar graphs and complete k-partite graphs, we give additional, more efficient constructions. We furthermore show that there are subdivisions of non-planar graphs that are not conflict graphs. Lastly, we show a separator lemma for conflict graphs

Maximum Clique in Disk-Like Intersection Graphs

International audienceWe study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude invoking the hardness of Maximum Independent Set on the class C, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks

Hydrophobization and Antimicrobial Activity of Chitosan and Paper-Based Packaging Material

This study reports the elaboration of water-resistant, antimicrobial, chitosan and paper-based materials as environmentally friendly food packaging materials. Two types of papers were coated with chitosan-palmitic acid emulsions or with a blend of chitosan and O,O'-dipalmitoylchitosan (DPCT). Micromorphology studies showed that inclusion of hydrophobic compounds into the chitosan matrix was enhanced by grafting them onto chitosan and that this led to their penetration of the paper's core. Compared to chitosan-coated papers, the coating of chitosan-palmitic emulsion kept vapor-barrier properties unchanged (239 and 170 g.m(-2).d(-1) versus 241 and 161 g.m(-2).d(-1)), while the coating of chitosan-DPCT emulsion dramatically deteriorated them (441 and 442 g.m(-2).d(-1)). However, contact angle measurements (110-120 degrees after 1 min) and penetration dynamics analysis showed that both strategies improved liquid-water resistance of the materials. Kit-test showed that all hydrophobized chitosan-coated papers kept good grease barrier properties (degree of resistance 6-8/12). Finally, all chitosan-coated materials exhibited over 98% inhibition on Salmonella Typhimurium and Listeria monocytogenes

Translation sur graphe avec préservation du voisinage

National audienceDans de nombreux domaines comme l'Internet des Objets ou la neuroimagerie, les signaux sont naturellement portés par des graphes. Ces derniers contiennent généralement des informations sur la similarité des valeurs de signaux observées aux différents sommets. L'un des intérêts à utiliser des graphes est qu'ils permettent de définir des opérateurs adaptés pour traiter les signaux qui y évoluent. Parmi ces opérateurs, un de prime importance pour de nombreux problèmes est la translation. Dans cet article, nous proposons de nouvelles définitions pour la translation sur graphe s'appuyant sur un nombre faible de propriétés simples. Plus précisément, nous proposons de définir ces translations comme des fonctions des sommets du graphe vers leurs voisins, préservant les relations de voisinage. Nous montrons que nos définitions, contrairement aux autres travaux sur le sujet, généralisent les translations usuelles sur des graphes grilles

Metropolitan Police Service No.3 Area North East

SIGLEAvailable from British Library Document Supply Centre-DSC:7498.810571(1996) / BLDSC - British Library Document Supply CentreGBUnited Kingdo