Approximating hitting sets of axis-parallel rectangles with opposite corners separated by a monotone curve
نویسندگان
چکیده
In this note, we present a simple combinatorial factor 8 algorithm for approximating the minimum hitting set of a family R = {R1, . . . ,Rn} of axis-parallel rectangles in the plane such that there exists an axis-monotone curve γ separating the same two opposite corners of each rectangle Ri (say, the lower left and upper right corners).
منابع مشابه
Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve
In this note, we present a simple combinatorial factor 6 algorithm for approximating the minimum hitting set of a family R = {R1, . . . ,Rn} of axis-parallel rectangles in the plane such that there exists an axis-monotone curve γ that intersects each rectangle in the family. The quality of the hitting set is shown by comparing it to the size of a packing (set of pairwise non-intersecting rectan...
متن کاملHitting and Piercing Rectangles Induced by a Point Set
We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number...
متن کاملOutput-Sensitive Algorithms for Enumerating Minimal Transversals for Some Geometric Hypergraphs
We give a general framework for the problem of finding all minimal hitting sets of a family of objects in R by another. We apply this framework to the following problems: (i) hitting hyper-rectangles by points in R; (ii) stabbing connected objects by axis-parallel hyperplanes in R; and (iii) hitting half-planes by points. For both the covering and hitting set versions, we obtain incremental pol...
متن کاملEmpty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on n points is shown to be ⌊ 1 4 n + n− 2⌋. This number also has other interpretations: • It is the maximum number of edges of a graph of dimension [3ll4], i.e., of a graph with a realizer of the form π1, π2, π1, π2....
متن کاملApproximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons
We investigate the problem of finding a minimum-area container for the disjoint packing of a set of convex polygons by translations. In particular, we consider axis-parallel rectangles or arbitrary convex sets as containers. For both optimization problems which are NP-hard we develop efficient constant factor approximation algorithms.
متن کامل