Submodular Function Maximization on the Bounded Integer Lattice
We consider the problem of maximizing a submodular function on the bounded integer lattice. As a direct generalization of submodular set functions, f:0,…,Cn→R+ $$f: \{0, \ldots , C\}^n \rightarrow \mathbb {R}_+$$ is submodular, if f(x)+f(y)≥f(x∧y)+f(x∨y) $$f(x) + f(y) \ge f(x \wedge y) + f(x \vee y)...
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| Published in | Approximation and Online Algorithms Vol. 9499; pp. 133 - 144 |
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| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
01.01.2015
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3319286838 9783319286839 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-319-28684-6_12 |
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| Summary: | We consider the problem of maximizing a submodular function on the bounded integer lattice. As a direct generalization of submodular set functions, f:0,…,Cn→R+ $$f: \{0, \ldots , C\}^n \rightarrow \mathbb {R}_+$$ is submodular, if f(x)+f(y)≥f(x∧y)+f(x∨y) $$f(x) + f(y) \ge f(x \wedge y) + f(x \vee y)$$ for all x,y∈0,…,Cn $$x,y \in \{0, \ldots , C\}^n$$ where ∧ $$\wedge $$ and ∨ $$\vee $$ denote element-wise minimum and maximum. The objective is finding a vector x maximizing f(x). In this paper, we present a deterministic 13 $$\frac{1}{3}$$ -approximation using a framework inspired by [2]. We also provide an example that shows the analysis is tight and yields additional insight into the possibilities of modifying the algorithm. Moreover, we examine some structural differences to maximization of submodular set functions which make our problem harder to solve. |
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| Bibliography: | Original Abstract: We consider the problem of maximizing a submodular function on the bounded integer lattice. As a direct generalization of submodular set functions, f:{0,…,C}n→R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: \{0, \ldots , C\}^n \rightarrow \mathbb {R}_+$$\end{document} is submodular, if f(x)+f(y)≥f(x∧y)+f(x∨y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x) + f(y) \ge f(x \wedge y) + f(x \vee y)$$\end{document} for all x,y∈{0,…,C}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y \in \{0, \ldots , C\}^n$$\end{document} where ∧\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\wedge $$\end{document} and ∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vee $$\end{document} denote element-wise minimum and maximum. The objective is finding a vector x maximizing f(x). In this paper, we present a deterministic 13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{3}$$\end{document}-approximation using a framework inspired by [2]. We also provide an example that shows the analysis is tight and yields additional insight into the possibilities of modifying the algorithm. Moreover, we examine some structural differences to maximization of submodular set functions which make our problem harder to solve. |
| ISBN: | 3319286838 9783319286839 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-319-28684-6_12 |