COCOON 2011: Computing and Combinatorics pp 331-342

# A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition

• Binh-Minh Bui-Xuan
• Pinar Heggernes
• Daniel Meister
• Andrzej Proskurowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

## 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.

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## Authors and Affiliations

• Binh-Minh Bui-Xuan
• 1
• Pinar Heggernes
• 1
• Daniel Meister
• 2
• Andrzej Proskurowski
• 3
1. 1.Department of InformaticsUniversity of BergenNorway
2. 2.Theoretical Computer ScienceUniversity of TrierGermany
3. 3.Department of Information and Computer ScienceUniversity of OregonUSA