Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-uniform Geometric Set Cover and Scheduling on Multiple Machines   We consider the following general scheduling problem studied recently by
Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size
$p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its
completion time. The goal is to find a schedule on $m$ machines with minimum
total cost. We give an $O(1)$ approximation for the problem, improving upon the
previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio),
and resolving the open question of Moseley.
  We first note that the scheduling problem can be reduced to a clean geometric
set cover problem where points on a line with arbitrary demands, must be
covered by a minimum cost collection of given intervals with non-uniform
capacity profiles. Unfortunately, current techniques for such problems based on
knapsack cover inequalities and low union complexity, completely lose the
geometric structure in the non-uniform capacity profiles and incur at least an
$\Omega(\log\log P)$ loss.
  To this end, we consider general covering problems with non-uniform
capacities, and give a new method to handle capacities in a way that completely
preserves their geometric structure. This allows us to use sophisticated
geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in
previous approaches. In addition to the scheduling problem above, we use this
approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic
capacitated covering problems.

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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph