Making Octants Colorful and Related Covering Decomposition Problems

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k colors so that every translate of the negative octant containing at least k^6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.Comment: version after revision process; minor changes in the expositio

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This can be interpreted as coloring point sets in R^2 with k colors such that any bottomless rectangle containing at least 3k-2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function $p(k)$ exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the European Workshop on Computational Geometry, held in Assisi (Italy) on March 19-21, 201

Proper Coloring of Geometric Hypergraphs

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions

Coloring Points with Respect to Squares

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane (Formula presented.) can be 2-colored such that every axis-parallel square that contains at least m points from (Formula presented.) contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram

Survey on Decomposition of Multiple Coverings

The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decompos-ability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions. 1 Cover-decomposability and the sensor cover problem Let P = { Pi | i ∈ I} be a collection of sets in Rd. We say that P is an m-fold covering if every point of Rd is contained in at least m members of P. The largest such m is called the thickness of the covering. A 1-fold covering is simply called a covering. To formulate the central question of this survey succinctly, we need a denition. Denition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings. Note that the above term is slightly misleading: we decompose (partition) not the set P, but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P