Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function f:2N→R+ $$f:2^{\mathcal {N}} \rightarrow \mathbb {R}^+$$ subject to a p-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and mo...

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Published inAutomata, Languages, and Programming Vol. 9134; pp. 318 - 330
Main Authors Chekuri, Chandra, Gupta, Shalmoli, Quanrud, Kent
Format Book Chapter
LanguageEnglish
Published Germany Springer Berlin / Heidelberg 01.01.2015
Springer Berlin Heidelberg
SeriesLecture Notes in Computer Science
Subjects
Algorithms & data structures
Cardinality Constraint
Greedy Algorithm
Knapsack Constraint
Network hardware
Oracle Access
Submodular Function
User interface design & usability
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Summary:We consider the problem of maximizing a nonnegative submodular set function f:2N→R+ $$f:2^{\mathcal {N}} \rightarrow \mathbb {R}^+$$ subject to a p-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are non-monotone. We describe deterministic and randomized algorithms that obtain a Ω(1p) $$\varOmega (\frac{1}{p})$$ -approximation using O(klogk) $$O(k \log k)$$ -space, where k is an upper bound on the cardinality of the desired set. The model assumes value oracle access to f and membership oracles for the matroids defining the p-matchoid constraint.
Bibliography:Original Abstract: We consider the problem of maximizing a nonnegative submodular set function f:2N→R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:2^{\mathcal {N}} \rightarrow \mathbb {R}^+$$\end{document} subject to a p-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are non-monotone. We describe deterministic and randomized algorithms that obtain a Ω(1p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (\frac{1}{p})$$\end{document}-approximation using O(klogk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(k \log k)$$\end{document}-space, where k is an upper bound on the cardinality of the desired set. The model assumes value oracle access to f and membership oracles for the matroids defining the p-matchoid constraint.
C. Chekuri—Work on this paper supported in part by NSF grant CCF-1319376.S. Gupta—Work on this paper supported in part by NSF grant CCF-1319376. K. Quanrud—Work on this paper supported in part by NSF grants CCF-1319376, CCF-1421231, and CCF-1217462.
ISBN:9783662476710
3662476711
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-662-47672-7_26