Lower bounds for three algorithms for transversal hypergraph generation

The computation of all minimal transversals of a given hypergraph in output-polynomial time is a long standing open question known as Transversal Hypergraph Generation. One of the first attempts at this problem—the sequential method [Claude Berge, Hypergraphs, in: North-Holland Mathematical Library, vol. 45, North-Holland, 1989]—is not output-polynomial as was shown by Takata [Ken Takata, A worst-case analysis of the sequential method to list the minimal hitting sets of a hypergraph, SIAM Journal on Discrete Mathematics 21 (4) (2007) 936–946]. Recently, three new algorithms improving the sequential method were published and experimentally shown to perform very well in practice [James Bailey, Thomas Manoukian, Kotagiri Ramamohanarao, A fast algorithm for computing hypergraph transversals and its application in mining emerging patterns, in: Proceedings of the 3rd IEEE International Conference on Data Mining, ICDM 2003, 19–22 December 2003, Melbourne, FL, USA, IEEE Computer Society, 2003, pp. 485–488; Guozhu Dong, Jinyan Li, Mining border descriptions of emerging patterns from dataset pairs, Knowledge and Information Systems 8 (2) (2005) 178–202; Dimitris J. Kavvadias, Elias C. Stavropoulos, An efficient algorithm for the transversal hypergraph generation, Journal of Graph Algorithms and Applications 9 (2) (2005) 239–264]. Nevertheless, a theoretical worst-case analysis has been pending. We close this gap by proving lower bounds for all three algorithms. Thereby, we show that none of them are output-polynomial.