A general algorithmic scheme for combinatorial decompositions with application to modular decompositions of hypergraphs

We study here algorithmic aspects of modular decomposition of hypergraphs. In the literature one can find three different definitions of modules: standard modules [25], k-subset modules [6], and Courcelle's modules [12]. Using fundamental tools from combinatorial decompositions such as partitive and orthogonal families, we directly derive a linear time algorithm for Courcelle's decomposition. We then introduce a general algorithmic tool for weakly partitive families and apply it for the three definitions of modules to derive polynomial time algorithms. For standard modules, we give an O(n3⋅l) time algorithm (where n is the number of vertices and l the sum of the size of the edges). For k-subset modules, we obtain an O(n3⋅l) runtime. This is an improvement from the best known algorithm for k-subset modular decomposition, which was not polynomial w.r.t. n and m, and is in O(n3k−5) time [6] where k denotes the maximal size of an edge, and m the number of hyperedges. Finally we focus on applications of orthogonality to modular decompositions of tournaments, simplifying the algorithm from [24]. The question of designing a O(n⋅l) time algorithm for the standard modular decomposition of hypergraphs remains open.