Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, with near-optimal results known under a variety of constraints when the submodular function is monotone. In this paper, we give constant approximation algorith...
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| Published in | Internet and Network Economics pp. 246 - 257 |
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| Main Authors | , , , |
| Format | Book Chapter |
| Language | English |
| Published |
Berlin, Heidelberg
Springer Berlin Heidelberg
2010
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| Series | Lecture Notes in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3642175716 9783642175718 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-642-17572-5_20 |
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| Summary: | Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, with near-optimal results known under a variety of constraints when the submodular function is monotone. In this paper, we give constant approximation algorithms for the non-monotone case that work for p-independence systems (which generalize constraints given by the intersection of p matroids that had been studied previously), where the running time is $\text{poly}(n,p)$ . Our algorithms and analyses are simple, and essentially reduce non-monotone maximization to multiple runs of the greedy algorithm previously used in the monotone case.
We extend these ideas to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting (where elements arrive one at a time in random order and the algorithm must make irrevocable decisions) subject to uniform matroid or a partition matroid constraint. Finally, we give an O(logk) approximation in the secretary setting subject to a general matroid constraint of rank k. |
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| Bibliography: | Original Abstract: Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, with near-optimal results known under a variety of constraints when the submodular function is monotone. In this paper, we give constant approximation algorithms for the non-monotone case that work for p-independence systems (which generalize constraints given by the intersection of p matroids that had been studied previously), where the running time is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\text{poly}(n,p)$\end{document}. Our algorithms and analyses are simple, and essentially reduce non-monotone maximization to multiple runs of the greedy algorithm previously used in the monotone case. We extend these ideas to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting (where elements arrive one at a time in random order and the algorithm must make irrevocable decisions) subject to uniform matroid or a partition matroid constraint. Finally, we give an O(logk) approximation in the secretary setting subject to a general matroid constraint of rank k. |
| ISBN: | 3642175716 9783642175718 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-642-17572-5_20 |