Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Online Square-into-Square Packing   In 1967, Moon and Moser proved a tight bound on the critical density of
squares in squares: any set of squares with a total area of at most 1/2 can be
packed into a unit square, which is tight. The proof requires full knowledge of
the set, as the algorithmic solution consists in sorting the objects by
decreasing size, and packing them greedily into shelves. Since then, the online
version of the problem has remained open; the best upper bound is still 1/2,
while the currently best lower bound is 1/3, due to Han et al. (2008). In this
paper, we present a new lower bound of 11/32, based on a dynamic shelf
allocation scheme, which may be interesting in itself. We also give results for
the closely related problem in which the size of the square container is not
fixed, but must be dynamically increased in order to ac- commodate online
sequences of objects. For this variant, we establish an upper bound of 3/7 for
the critical density, and a lower bound of 1/8. When aiming for accommodating
an online sequence of squares, this corresponds to a 2.82...- competitive
method for minimizing the required container size, and a lower bound of 1.33 .
. . for the achievable factor.

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The categories of this paper are: math
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin