Faster Algorithms to Enumerate Hypergraph Transversals

  • Manfred Cochefert
  • Jean-François Couturier
  • Serge GaspersEmail author
  • Dieter Kratsch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time polynomial-space algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed \(k\ge 3\), our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximum number of minimal transversals in n-vertex hypergraphs of rank \(k\ge 3\). Our main algorithm is a branching algorithm whose running time is analyzed with Measure and Conquer. It enumerates all minimal transversals of hypergraphs of rank 3 in time \(O(1.6755^n)\). Our enumeration algorithms improve upon the best known algorithms for counting minimum transversals in hypergraphs of rank k for \(k\ge 3\) and for computing a minimum transversal in hypergraphs of rank k for \(k\ge 6\).



We thank Fabrizio Grandoni for initial discussions on this research. Dieter Kratsch acknowledges support from the French Research Agency, project GraphEn (ANR-15-CE40-0009). Serge Gaspers is the recipient of an Australian Research Council (ARC) Future Fellowship (project FT140100048) and acknowledges support under the ARC’s Discovery Projects funding scheme (project DP150101134). NICTA is funded by the Australian Government through the Department of Communications and the ARC through the ICT Centre of Excellence Program.


  1. 1.
    Cochefert, M., Couturier, J.-F., Gaspers, S., Kratsch, D.: Faster algorithms to enumerate hypergraph transversals. Technical report arxiv:1510.05093 (2015)
  2. 2.
    Couturier, J.-F., Heggernes, P., van ’t Hof, P., Kratsch, D.: Minimal dominating sets in graph classes: combinatorial bounds and enumeration. Theor. Comput. Sci. 487(8), 2–94 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNF–SAT. In: Proceedings of CCC, pp. 74–84 (2012)Google Scholar
  4. 4.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32(2), 514–537 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Elbassioni, K.M., Rauf, I.: Polynomial-time dualization of r-exact hypergraphs with applications in geometry. Discrete Math. 310(17–18), 2356–2363 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fernau, H.: Parameterized algorithmics for d-hitting set. Int. J. Comput. Math. 87(14), 3157–3174 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernau, H.: Parameterized algorithms for d-hitting set: the weighted case. Theor. Comput. Sci. 411(16–18), 1698–1713 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-hitting set. Algorithmica 57(1), 97–118 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fomin, F.V., Gaspers, S., Kratsch, D., Liedloff, M., Saurabh, S.: Iterative compression and exact algorithms. Theor. Comput. Sci. 411(7–9), 1045–1053 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. Technical report arxiv:1512.01621 (2015)
  12. 12.
    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gaspers, S.: Algorithmes exponentiels. Master’s thesis, University Metz, France (2005)Google Scholar
  15. 15.
    Gaspers, S., Algorithms, E.T.: Exponential Time Algorithms: Structures, Measures, and Bounds. VDM Verlag Dr. Mueller e.K, Saarbrücken (2010)Google Scholar
  16. 16.
    Gaspers, S., Sorkin, G.B.: A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between. J. Comput. Syst. Sci. 78(1), 305–335 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Golovach, P.A., Heggernes, P., Kratsch, D., Villanger, Y.: An incremental polynomial time algorithm to enumerate all minimal edge dominating sets. Algorithmica 72(3), 836–859 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the enumeration of minimal dominating sets and related notions. SIAM J. Discrete Math. 28(4), 1916–1929 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: A polynomial delay algorithm for enumerating minimal dominating sets in chordal graphs. In: Proceedings of WG (2015)Google Scholar
  20. 20.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 446–457. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  21. 21.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of computer computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  22. 22.
    Kavvadias, D.J., Stavropoulos, E.C.: An efficient algorithm for the transversal hypergraph generation. J. Graph Algorithms Appl. 9(2), 239–264 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khachiyan, L., Boros, E., Elbassioni, K.M., Gurvich, V.: On the dualization of hypergraphs with bounded edge-intersections and other related classes of hypergraphs. Theor. Comput. Sci. 382(2), 139–150 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Miller, R.E., Muller, D.E.: A problem of maximum consistent subsets. IBM Research Report RC-240, J. T. Watson Research Center (1960)Google Scholar
  25. 25.
    Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms 1(1), 89–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wahlström, M.: Exact algorithms for finding minimum transversals in rank-3 hypergraphs. J. Algorithms 51(2), 107–121 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Linköping University, Sweden (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Manfred Cochefert
    • 1
  • Jean-François Couturier
    • 2
  • Serge Gaspers
    • 3
    • 4
    Email author
  • Dieter Kratsch
    • 1
  1. 1.LITAUniversité de LorraineMetzFrance
  2. 2.CReSTICUniversité de ReimsReimsFrance
  3. 3.University of New South WalesSydneyAustralia
  4. 4.Data61 (formerly: NICTA)CSIROSydneyAustralia

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