A generic approach to decomposition algorithms, with an application to digraph decomposition ⋆ Binh-Minh Bui-Xuan 1 , Pinar Heggernes 1 , Daniel Meister 2 , and Andrzej Proskurowski 3 1 Department of Informatics, University of Bergen, Norway, buixuan@ii.uib.no, buixuan@lip6.fr, pinar.heggernes@ii.uib.no 2 Theoretical Computer Science, University of Trier, Germany, daniel.meister@uni-trier.de 3 Department of Information and Computer Science, University of Oregon, USA, andrzej@cs.uoregon.edu 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 rep- resentation of its members by labeled trees. The size of the tree is pro- portional 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 no- tion 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 decom- position. Finally, we give efficient algorithms for the two basic problems and obtain an O(n 3 )-time algorithm for computing a tree representation. 1 Introduction The running time of an algorithm that finds a solution by exhaustive search over the family of possible solutions is dependent on the number of possible solutions. For most practical applications, the number of possible solutions is large compared to the input size itself, which makes this brute-force algorithm very inefficient. If the family of possible solutions has some structure, an efficient and compact representation of the family may be a key step to designing an ⋆ This work was supported by the Research Council of Norway. The first author was supported by the French National Research Agency, project MAGNUM.