Parametric Streaming Two-Stage Submodular Maximization
We study the submodular maximization problem in generalized streaming setting using a two-stage policy. In the streaming context, elements are released in a fashion that an element is revealed at one time. Subject to a limited memory capacity, the problem aims to sieve a subset of elements with a su...
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| Published in | Theory and Applications of Models of Computation Vol. 12337; pp. 193 - 204 |
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| Main Authors | , , , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2020
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3030592669 9783030592660 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-030-59267-7_17 |
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| Summary: | We study the submodular maximization problem in generalized streaming setting using a two-stage policy. In the streaming context, elements are released in a fashion that an element is revealed at one time. Subject to a limited memory capacity, the problem aims to sieve a subset of elements with a sublinear size $$\ell $$ , such that the expecting objective value of all utility functions over the summarized subsets has a performance guarantee. We present a generalized one pass, $$\left( \gamma ^5_{\min }/(5+ 2\gamma ^2_{\min } )-O(\epsilon )\right) $$ -approximation, which consumes $$O(\epsilon ^{-1}\ell \log (\ell \gamma _{\min }^{-1}))$$ memory and runs in $$O(\epsilon ^{-1}kmn\log (\ell \gamma _{\min }^{-1}))$$ time, where k, n, m and $$\gamma _{\min }$$ denote the cardinality constraint, the element stream size, the amount of the learned functions, and the minimum generic submodular ratio of the learned functions, respectively. |
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| Bibliography: | Original Abstract: We study the submodular maximization problem in generalized streaming setting using a two-stage policy. In the streaming context, elements are released in a fashion that an element is revealed at one time. Subject to a limited memory capacity, the problem aims to sieve a subset of elements with a sublinear size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}, such that the expecting objective value of all utility functions over the summarized subsets has a performance guarantee. We present a generalized one pass, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \gamma ^5_{\min }/(5+ 2\gamma ^2_{\min } )-O(\epsilon )\right) $$\end{document}-approximation, which consumes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon ^{-1}\ell \log (\ell \gamma _{\min }^{-1}))$$\end{document} memory and runs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon ^{-1}kmn\log (\ell \gamma _{\min }^{-1}))$$\end{document} time, where k, n, m and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{\min }$$\end{document} denote the cardinality constraint, the element stream size, the amount of the learned functions, and the minimum generic submodular ratio of the learned functions, respectively. |
| ISBN: | 3030592669 9783030592660 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-030-59267-7_17 |