A More Compact Translation of Pseudo-Boolean Constraints into CNF Such That Generalized Arc Consistency Is Maintained

In this paper we answer the open question for the existence of a more compact encoding from Pseudo-Boolean constraints into CNF that maintains generalized arc consistency by unit propagation, formalized by Bailleux et al. in [21]. In contrast to other encodings our approach is defined in an abstract...

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Bibliographic Details
Published inKI 2014: Advances in Artificial Intelligence pp. 123 - 134
Main Authors Manthey, Norbert, Philipp, Tobias, Steinke, Peter
Format Book Chapter
LanguageEnglish
Published Cham Springer International Publishing 2014
SeriesLecture Notes in Computer Science
Subjects
Auxiliary Variable
Conjunctive Normal Form
Constraint Programming
Propositional Variable
Unit Propagation
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ISBN3319112058
9783319112053
ISSN0302-9743
1611-3349
DOI10.1007/978-3-319-11206-0_13

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Summary:In this paper we answer the open question for the existence of a more compact encoding from Pseudo-Boolean constraints into CNF that maintains generalized arc consistency by unit propagation, formalized by Bailleux et al. in [21]. In contrast to other encodings our approach is defined in an abstract way and we present a concrete instantiation, resulting in a space complexity of $\mathcal{O}(n^2 \text{\,log}^2(n)\text{\,log}(w_{\mathsf{max}}))$ clauses in contrast to $\mathcal{O}(n^3 \text{\,log}(n)\text{\,log}(w_{\mathsf{max}}))$ clauses generated by the previously best known encoding that maintains generalized arc consistency.
Bibliography:Original Abstract: In this paper we answer the open question for the existence of a more compact encoding from Pseudo-Boolean constraints into CNF that maintains generalized arc consistency by unit propagation, formalized by Bailleux et al. in [21]. In contrast to other encodings our approach is defined in an abstract way and we present a concrete instantiation, resulting in a space complexity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n^2 \text{\,log}^2(n)\text{\,log}(w_{\mathsf{max}}))$\end{document} clauses in contrast to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n^3 \text{\,log}(n)\text{\,log}(w_{\mathsf{max}}))$\end{document} clauses generated by the previously best known encoding that maintains generalized arc consistency.
ISBN:3319112058
9783319112053
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-11206-0_13