No file available
To read the file of this research,
you can request a copy directly from the authors.
Packing, Hitting and Coloring Squares
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
Given a family of squares in the plane, their asks for the maximum number, , of pairwise disjoint squares among them, while their asks for the minimum number, , of points hitting all of them, . Both problems are NP-hard even if all the rectangles are unit squares and their sides are parallel to the axes. The main results of this work are providing the first bounds for the ratio on not necessarily axis-parallel squares. We establish an upper bound of 6 for unit squares and 10 for squares of varying sizes. The worst ratios we can provide with examples are 3 and 4, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval for unit squares and for arbitrary squares. The new bounds necessitate a mixture of novel and classical techniques of possibly extendable use. Furthermore, we study rectangles with a bounded ``aspect ratio'', where the of a rectangle is the larger side of a rectangle divided by its smaller side. We improve on the well-known best bound, which is quadratic in terms of the aspect ratio. We reduce it from quadratic to linear for rectangles, even if they are not axis-parallel, and from linear to logarithmic, for axis-parallel rectangles. Finally, we prove similar bounds for the chromatic numbers of squares and rectangles with a bounded aspect ratio.
ResearchGate has not been able to resolve any citations for this publication.
Finding a maximum independent set (MIS) of a given fam- ily of
axis-parallel rectangles is a basic problem in computational geom- etry
and combinatorics. This problem has attracted signi?cant atten- tion
since the sixties, when Wegner conjectured that the corresponding
duality gap, i.e., the maximum possible ratio between the maximum
independent set and the minimum hitting set (MHS), is bounded by a
universal constant. An interesting special case, that may prove use- ful
to tackling the general problem, is the diagonal-intersecting case, in
which the given family of rectangles is intersected by a diagonal.
Indeed, Chepoi and Felsner recently gave a factor 6 approximation
algorithm for MHS in this setting, and showed that the duality gap is
between 3/2 and 6. In this paper we improve upon these results. First we
show that MIS in diagonal-intersecting families is NP-complete,
providing one smallest subclass for which MIS is provably hard. Then, we
derive an -time algorithm for the maximum weight independent set
when, in addition the rectangles intersect below the diagonal. This
improves and extends a classic result of Lubiw, and amounts to obtain a
2-approximation algo- rithm for the maximum weight independent set of
rectangles intersecting a diagonal. Finally, we prove that for
diagonal-intersecting families the duality gap is between 2 and 4. The
upper bound, which implies an approximation algorithm of the same
factor, follows from a simple com- binatorial argument, while the lower
bound represents the best known lower bound on the duality gap, even in
the general case.
In this paper we show that the chromatic number of intersection graphs of congruent geometric figures obtained by translations of a fixed figure in the plane is bounded by the clique number. Further, in the paper we prove that the triangle-free intersection graph of a finite number of compact connected sets with piecewise differentiable Jordan curve boundaries is planar and hence, is 3-colorable.
The problem of the partition-numbersJ
ℱ(p, q), considered by Hadwiger and Debrunner for the family ℱ=C
n of convex bodies, is extended to simplicial complexes and arbitrary families assuming only the validity of Helly’s theorem. We obtain results similar to those of Hadwiger and Debrunner. Further we show the existence of all partition-numbers for the familyℱ = H
nC of homothets of a convex body and we get new informations on the partition-numbers for the family of parallel rectangles.
An intersection graph of rectangles in the (x, y)-plane with sides parallel to the axes is obtained by representing each rectangle by a vertex and connecting two vertices by an edge if and only if the corresponding rectangles intersect. This paper describes algorithms for two problems on intersection graphs of rectangles in the plane. One is an O(n log n) algorithm for finding the connected components of an intersection graph of n rectangles. This algorithm is optimal to within a constant factor. The other is an O(n log n) algorithm for finding a maximum clique of such a graph. It seems interesting that the maximum clique problem is polynomially solvable, because other related problems, such as the maximum stable set problem and the minimum clique cover problem, are known to be NP-complete for intersection graphs of rectangles. Furthermore, we briefly show that the k-colorability problem on intersection graphs of rectangles is NP-complete.
We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.
It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of Simple Max Cut (Max Cut with edge weights restricted to value 1), and, as a corollary, the completeness of the Optimal Linear Arrangement problem. We then show that even if the domains of the Node Cover and Directed Hamiltonian Path problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For Graph 3-Colorability, Node Cover, and Undirected Hamiltonian Circuit, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.
Let G be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number k is (3k 3)-degenerate. This bound is sharp. As a consequence, we derive that G is (3k 2)-colorable. We show also that the chromatic number of every intersection graph H of a family of homothetic copies of a fixed convex set in the plane with clique number k is at most 6k 6.
The problem of embedding the interconnection pattern of a circuit into a two-dimensional surface of minimal area is discussed. Since even for some natural patterns graphs containing m connections may require Ω(m2) area, in order to achieve compact embeddings restricted classes of graphs have to be considered. For example, arbitrary trees (of bounded degree) can be embedded in linear area without edges crossing over. Planar graphs can be embedded efficiently only if crossovers are allowed in the embedding.
. In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal-sized
disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving
on a result of Clark et al. [2] it is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k≥ 3 . Furthermore, a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques.
We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n. Comment: 11 pages, 2 figures
According to a classical result of Grünbaum, the transversal number of any family of pairwise-intersecting translates or homothets of a convex body C in ℝd
is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number to the packing number over all families of translates (resp. homothets) of a convex body C in ℝd
. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝd
, and gave the first bounds on α(C) for convex bodies C in ℝd
and on β(C) for convex bodies C in the plane. In this paper, we show that β(C) is also bounded by a function of d for any convex body C in ℝd
, and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝd
for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.
Let F be a family of translates of a fixed convex set M in Rn. Let (F) and (F) denote the transversal number and the independence number of F, respectively. We show that (F)(F) 8(F) − 5 for n = 2 and (F) 2n−1nn(F) for n 3. Furthermore, if M is centrally symmetric convex body in the plane, then (F)(F) 6(F) − 3. © 2006 Elsevier B.V. All rights reserved.
: A computational method for finding good coverings of a square with equal circles is presented. The algorithm uses a quasi-Newton method with BFGS secant update to minimize the uncovered area by moving the circles. The radius of the circles is further adapted to find locally optimal coverings, and the algorithm is applied repeatedly to random initial configurations. The structures of the coverings are determined and the coordinates of each circle are calculated with high precision using a mathematical model for an idealized physical structure consisting of tensioned bars and frictionless pin joints. Best found coverings of a square with up to 30 circles are displayed, 17 of which are new or improve on earlier published coverings. KEYWORDS: circle covering, equidistant graph, square CONTENTS 1 Introduction 1 2 The Covering Algorithm 1 3 Obtaining Numerical Solutions 4 4 The Results 5 1 INTRODUCTION Packing circles in various shapes in the plane or on a sphere has been a popular res...
Intersecting graphs of rectangles and segments
- R Ahlswede
- I Karapetyan
R. Ahlswede and I. Karapetyan. "Intersecting graphs of rectangles and segments". In: LNCS
4123 (2006), pp. 1064-1065.
Simple Linear Time Algorithms For Piercing Pairwise Intersecting Disks
- A Biniaz
- P Bose
- Y Wang
A. Biniaz, P. Bose, and Y. Wang. "Simple Linear Time Algorithms For Piercing Pairwise
Intersecting Disks". In: Proceedings of the 33rd Canadian Conference on Computational Geometry. CCCG 2021. Dalhousie University Halifax, Nova Scotia Canada, 2021, pp. 228-236.
Piercing axis-parallel boxes
- M Chudnovsky
- S Sprinkl
- S Zerbib
M. Chudnovsky, S. Sprinkl, and S. Zerbib. "Piercing axis-parallel boxes". In: The Electronic
Journal of Combinatorics 25(1) (2018), P1.70.
1. The k-colorability of axis-parallel unit squares is NP-complete for any fixed k ∈ N, k ≥ 3, and so is the existence of an independent set of unit squares larger than a given size
- A Theorem
Theorem A.1. The k-colorability of axis-parallel unit squares is NP-complete for any fixed k ∈
N, k ≥ 3, and so is the existence of an independent set of unit squares larger than a given size.