The one-round Voronoi game replayed

We consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough $n$ and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.Comment: 14 pages, 6 figures, Latex; revised for journal version, to appear in Computational Geometry: Theory and Applications. Extended abstract version appeared in Workshop on Algorithms and Data Structures, Springer Lecture Notes in Computer Science, vol.2748, 2003, pp. 150-16

Graphes et structures algébriques associées, labyrinthes, cartes, réseaux d’automates, systèmes acentrés

Pierre Rosenstiehl, directeur d’études Combinatoire et graphes. Taxiplanie Les ordinateurs graphiques permettent de mieux en mieux de multiplier les visualisations dynamiques, d’expérimenter en trois dimensions, ce qui conduit à formuler des conjectures auparavant inaccessibles. On en est ainsi venu, non seulement à maîtriser topologiquement des objets complexes plongés dans le plan, mais aussi à savoir les déformer algorithmiquement, et de façon efficace, tout en leur imprimant des contraint..

Claude Berge, ses graphes et hypergraphes

Tribute to Claude BergeHommage du CAMS à Claude Berg

Domino Tatami Covering is NP-complete

A covering with dominoes of a rectilinear region is called \emph{tatami} if no four dominoes meet at any point. We describe a reduction from planar 3SAT to Domino Tatami Covering. As a consequence it is NP-complete to decide whether there is a perfect matching of a graph that meets every 4-cycle, even if the graph is restricted to be an induced subgraph of the grid-graph. The gadgets used in the reduction were discovered with the help of a SAT-solver.Comment: 10 pages, accepted at The International Workshop on Combinatorial Algorithms (IWOCA) 201

Graphes et structures algébriques associées, labyrinthes, réseaux d’automates, systèmes acentrés

Pierre Rosenstiehl, directeur d’études Combinatoire et graphes. Taxiplanie Dans le plan, l’étude d’objets géométriques déformables nécessite des concepts algébriques adaptés : corps finis, Z-modules, matroïdes, polynômes énumérateurs, permutations, arbres et treillis, codes parenthésés. Les ordinateurs graphiques permettent de mieux en mieux de multiplier les visualisations dynamiques, d’expérimenter en trois dimensions, ce qui conduit à formuler des conjectures auparavant inaccessibles. On e..

Graphes et structures algébriques associées, labyrinthes, réseaux d’automates, systèmes acentrés

Pierre Rosenstiehl, directeur d’études Nous avons poursuivi et discuté en séminaire notre étude d’un nouveau code représentant cartes et hypercartes par une seule permutation. Ces résultats conjecturés dans diverses thèses de combinatoire apportent un éclairage nouveau sur la description et l’exploration de figures du plan (avec P. Ossona de Mendez). L’étude d’objets géométriques finis et déformables du plan nécessite des concepts algébriques adaptés : corps finis, Z-modules, matroïdes, polyn..

Sorting Jordan sequences in linear time

For a Jordan curve C in the plane, let x_{1},x_{2},...,x_{n} be the abscissas of the intersection points of C with the x-axis, listed in the order the points occur on C. We call x_{1},x_{2},...,x_{n} a Jordan sequence. In this paper we describe an O(n)-time algorithm for recognizing and sorting Jordan sequences. The problem of sorting such sequences arises in computational geometry and computational geography. Our algorithm is based on a reduction of the recognition and sorting problem to a list-splitting problem. To solve the list-splitting problem we use level linked search trees

Marc Barbut, le formateur universel

Marc Barbut, rassembleur et formateur, réussit en franchissant les cloisonnements institutionnels hexagonaux l’implantation de programmes de mathématique adaptés aux étudiants en sciences humaines à tous les niveaux.A talented teacher, Marc Barbut, successfully trained and brought together colleagues from many different institutions. He launched mathematical programs of every level suited to the needs of the new student populations in social sciences

Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

La question de la modélisation en sciences humaines : mathématiques et informatique

Henri Berestycki, Pierre Rosenstiehl, Jean Petitot, directeurs d’études La question de la modélisation en sciences humaines : mathématiques et informatique Ce séminaire a été consacré à la présentation de développements récents de la modélisation mathématique dans des domaines variés des sciences humaines. Les exposés ont été prolongés par des débats sur le statut, le rôle et la pertinence des modèles mathématiques dans des domaines assez fortement formalisés des sciences humaines et sociales..