Volume 6842 of the series Lecture Notes in Computer Science pp 331342
A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition
 BinhMinh BuiXuanAffiliated withDepartment of Informatics, University of Bergen
 , Pinar HeggernesAffiliated withDepartment of Informatics, University of Bergen
 , Daniel MeisterAffiliated withTheoretical Computer Science, University of Trier
 , Andrzej ProskurowskiAffiliated withDepartment of Information and Computer Science, University of Oregon
Abstract
A set family is a collection of sets over a universe. If a set family satisfies certain closure properties then it admits an efficient representation of its members by labeled trees. The size of the tree is proportional to the size of the universe, whereas the number of set family members can be exponential. Computing such efficient representations is an important task in algorithm design. Set families are usually not given explicitly (by listing their members) but represented implicitly.
We consider the problem of efficiently computing tree representations of set families. Assuming the existence of efficient algorithms for solving the Membership and Separation problems, we prove that if a set family satisfies weak closure properties then there exists an efficient algorithm for computing a tree representation of the set family. The running time of the algorithm will mainly depend on the running times of the algorithms for the two basic problems. Our algorithm generalizes several previous results and provides a unified approach to the computation for a large class of decompositions of graphs. We also introduce a decomposition notion for directed graphs which has no undirected analogue. We show that the results of the first part of the paper are applicable to this new decomposition. Finally, we give efficient algorithms for the two basic problems and obtain an \({\cal O}(n^3)\)time algorithm for computing a tree representation.
 Title
 A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition
 Book Title
 Computing and Combinatorics
 Book Subtitle
 17th Annual International Conference, COCOON 2011, Dallas, TX, USA, August 1416, 2011. Proceedings
 Pages
 pp 331342
 Copyright
 2011
 DOI
 10.1007/9783642226854_30
 Print ISBN
 9783642226847
 Online ISBN
 9783642226854
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 6842
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Bin Fu ^{(16)}
 DingZhu Du ^{(17)}
 Editor Affiliations

 16. Department of Computer Science, University of TexasPan American
 17. Department of Computer Science, University of Texas at Dallas
 Authors

 BinhMinh BuiXuan ^{(18)}
 Pinar Heggernes ^{(18)}
 Daniel Meister ^{(19)}
 Andrzej Proskurowski ^{(20)}
 Author Affiliations

 18. Department of Informatics, University of Bergen, Norway
 19. Theoretical Computer Science, University of Trier, Germany
 20. Department of Information and Computer Science, University of Oregon, USA
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