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Faster Algorithms to Enumerate Hypergraph Transversals

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LATIN 2016: Theoretical Informatics (LATIN 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9644))

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Abstract

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 \(k\ge 6\).

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References

  1. Cochefert, M., Couturier, J.-F., Gaspers, S., Kratsch, D.: Faster algorithms to enumerate hypergraph transversals. Technical report arxiv:1510.05093 (2015)

  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)

    MathSciNet  MATH  Google Scholar 

  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. Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32(2), 514–537 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fernau, H.: Parameterized algorithmics for d-hitting set. Int. J. Comput. Math. 87(14), 3157–3174 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fernau, H.: Parameterized algorithms for d-hitting set: the weighted case. Theor. Comput. Sci. 411(16–18), 1698–1713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-hitting set. Algorithmica 57(1), 97–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. Technical report arxiv:1512.01621 (2015)

  12. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  13. Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gaspers, S.: Algorithmes exponentiels. Master’s thesis, University Metz, France (2005)

    Google Scholar 

  15. Gaspers, S., Algorithms, E.T.: Exponential Time Algorithms: Structures, Measures, and Bounds. VDM Verlag Dr. Mueller e.K, Saarbrücken (2010)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  22. Kavvadias, D.J., Stavropoulos, E.C.: An efficient algorithm for the transversal hypergraph generation. J. Graph Algorithms Appl. 9(2), 239–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  26. Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms 1(1), 89–102 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wahlström, M.: Exact algorithms for finding minimum transversals in rank-3 hypergraphs. J. Algorithms 51(2), 107–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Linköping University, Sweden (2007)

    Google Scholar 

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Acknowledgments

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.

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Authors and Affiliations

  1. LITA, Université de Lorraine, Metz, France

    Manfred Cochefert & Dieter Kratsch

  2. CReSTIC, Université de Reims, Reims, France

    Jean-François Couturier

  3. University of New South Wales, Sydney, Australia

    Serge Gaspers

  4. Data61 (formerly: NICTA), CSIRO, Sydney, Australia

    Serge Gaspers

Authors
  1. Manfred Cochefert
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  2. Jean-François Couturier
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  3. Serge Gaspers
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  4. Dieter Kratsch
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Corresponding author

Correspondence to Serge Gaspers .

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, ON, Ontario, Canada

    Evangelos Kranakis

  2. University Chile, Santiago, Chile

    Gonzalo Navarro

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Cochefert, M., Couturier, JF., Gaspers, S., Kratsch, D. (2016). Faster Algorithms to Enumerate Hypergraph Transversals. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_23

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_23

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-49528-5

  • Online ISBN: 978-3-662-49529-2

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