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Approximations for Monotone and Non-monotone Submodular Maximization with Knapsack Constraints
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Ariel Kulik, M.Sc. Thesis Seminar
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Wednesday, 15.12.2010, 16:30
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Taub 601
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. 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. Our approach has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and Maximum Coverage with additional knapsack constraints.
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