2,566 research outputs found
Packing Disks into Disks with Optimal Worst-Case Density
We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area delta 0 there are sets of disks of area 1/2+epsilon that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms
Worst-Case Optimal Covering of Rectangles by Disks
We provide the solution for a fundamental problem of geometric optimization
by giving a complete characterization of worst-case optimal disk coverings of
rectangles: For any , the critical covering area
is the minimum value for which any set of disks with total area at least
can cover a rectangle of dimensions .
We show that there is a threshold value , such that for the critical
covering area is , and for , the critical area is ; these
values are tight.
For the special case , i.e., for covering a unit square, the
critical covering area is . The proof
uses a careful combination of manual and automatic analysis, demonstrating the
power of the employed interval arithmetic technique.Comment: 45 pages, 26 figures. Full version of an extended abstract with the
same title accepted for publication in the proceedings of the 36th Symposium
on Computational Geometry (SoCG 2020
Packing Squares into a Disk with Optimal Worst-Case Density
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is ? = 8/(5?)? 0.509. This implies that any set of (not necessarily equal) squares of total area A ? 8/5 can always be packed into a disk with radius 1; in contrast, for any ? > 0 there are sets of squares of total area 8/5+? that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square (1/2), circles in a square (?/(3+2?2) ? 0.539) and circles in a circle (1/2) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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