13,671 research outputs found
Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio
We use computational experiments to find the rectangles of minimum area into
which a given number n of non-overlapping congruent circles can be packed. No
assumption is made on the shape of the rectangles. Most of the packings found
have the usual regular square or hexagonal pattern. However, for 1495 values of
n in the tested range n =< 5000, specifically, for n = 49, 61, 79, 97, 107,...
4999, we prove that the optimum cannot possibly be achieved by such regular
arrangements. The evidence suggests that the limiting height-to-width ratio of
rectangles containing an optimal hexagonal packing of circles tends to
2-sqrt(3) as n tends to infinity, if the limit exists.Comment: 21 pages, 13 figure
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
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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Packing items from a triangular distribution
We consider the problem of packing n items which are drawn according to a probability distribution whose density function is triangular in shape. For triangles which represent density functions whose expectation is 1/p for p = 3, 4, 5, ..., we give a packing strategy for which the ratio of the number of bins used in the packing to the expected total size of the items asymptotically approaches 1
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