Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Approximations for Monotone and Non-monotone Submodular Maximization
  with Knapsack Constraints   Submodular maximization generalizes many fundamental problems in discrete
optimization, including Max-Cut in directed/undirected graphs, maximum
coverage, maximum facility location and marketing over social networks.
  In this paper we consider the problem of maximizing any submodular function
subject to $d$ knapsack constraints, where $d$ is a fixed constant. We
establish a strong relation between the discrete problem and its continuous
relaxation, obtained through {\em extension by expectation} of the submodular
function. Formally, we show that, for any non-negative submodular function, an
$\alpha$-approximation algorithm for the continuous relaxation implies a
randomized $(\alpha - \eps)$-approximation algorithm for the discrete problem.
We use this relation to improve the best known approximation ratio for the
problem to $1/4- \eps$, for any $\eps > 0$, and to obtain a nearly optimal
$(1-e^{-1}-\eps)-$approximation ratio for the monotone case, for any $\eps>0$.
We further show that the probabilistic domain defined by a continuous solution
can be reduced to yield a polynomial size domain, given an oracle for the
extension by expectation. This leads to a deterministic version of our
technique.

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The categories of this paper are: stat
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio