Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Revisiting Modified Greedy Algorithm for Monotone Submodular
  Maximization with a Knapsack Constraint   Monotone submodular maximization with a knapsack constraint is NP-hard.
Various approximation algorithms have been devised to address this optimization
problem. In this paper, we revisit the widely known modified greedy algorithm.
First, we show that this algorithm can achieve an approximation factor of
$0.405$, which significantly improves the known factors of $0.357$ given by
Wolsey and $(1-1/\mathrm{e})/2\approx 0.316$ given by Khuller et al. More
importantly, our analysis closes a gap in Khuller et al.'s proof for the
extensively mentioned approximation factor of $(1-1/\sqrt{\mathrm{e}})\approx
0.393$ in the literature to clarify a long-standing misconception on this
issue. Second, we enhance the modified greedy algorithm to derive a
data-dependent upper bound on the optimum. We empirically demonstrate the
tightness of our upper bound with a real-world application. The bound enables
us to obtain a data-dependent ratio typically much higher than $0.405$ between
the solution value of the modified greedy algorithm and the optimum. It can
also be used to significantly improve the efficiency of algorithms such as
branch and bound.

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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin