IWOCA 2019: Combinatorial Algorithms pp 251-264

# A General Algorithmic Scheme for Modular Decompositions of Hypergraphs and Applications

• Michel Habib
• Fabien de Montgolfier
• Mengchuan Zou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11638)

## Abstract

We study here algorithmic aspects of modular decomposition of hypergraphs. In the literature one can find three different definitions of modules, namely: the standard one [19], the k-subset modules [6] and the Courcelle’s one [11]. Using the fundamental tools defined for combinatorial decompositions such as partitive and orthogonal families, we directly derive a linear time algorithm for Courcelle’s decomposition. Then we introduce a general algorithmic tool for partitive families and apply it for the other two definitions of modules to derive polynomial algorithms. For standard modules it leads to an algorithm in $$O(n^3 \cdot l)$$ time (where n is the number of vertices and l is the sum of the size of the edges). For k-subset modules we obtain $$O(n^3\cdot m\cdot l)$$ (where m is the number of edges). This is an improvement from the best known algorithms for k-subset modular decomposition, which was not polynomial w.r.t. n and m, and is in $$O(n^{3k-5})$$ time [6] where k denotes the maximal size of an edge. Finally we focus on applications of orthogonality to modular decompositions of tournaments, simplifying the algorithm from [18]. The question of designing a linear time algorithms for the standard modular decomposition of hypergraphs remains open.

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## Authors and Affiliations

• Michel Habib
• 1
• 3
Email author
• Fabien de Montgolfier
• 1
• 3