Incremental Complexity of a Bi-objective Hypergraph Transversal Problem

  • Ricardo Andrade
  • Etienne Birmelé
  • Arnaud Mary
  • Thomas Picchetti
  • Marie-France Sagot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)

Abstract

The hypergraph transversal problem has been intensively studied, both from a theoretical and a practical point of view. In particular, its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs \((\mathcal {A},\mathcal {B})\), and the aim is to enumerate minimal sets which hit all the hyperedges of \(\mathcal {A}\) while intersecting a minimal set of hyperedges of \(\mathcal {B}\). In this paper, we formalize this problem and relate it to the enumeration of minimal hitting sets of bundles. We show cases when under degree or dimension contraints these problems remain NP-hard, and give a polynomial algorithm for the case when \(\mathcal {A}\) has bounded dimension, by building a hypergraph whose transversals are exactly the hitting sets of bundles.

References

  1. 1.
    Angel, E., Bampis, E., Gourvès, L.: On the minimum hitting set of bundles problem. Theoret. Comput. Sci. 410(45), 4534–4542 (2009)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North-Holland, Amsterdam (1989)MATHGoogle Scholar
  3. 3.
    Bertrand, D., Chng, K.R., Sherbaf, F.G., Kiesel, A., Chia, B.K.H., Sia, Y.Y., Huang, S.K., Hoon, D.S.B., Liu, T., Hillmer, A., Hillmer, A., Nagarajan, N.: Patient-specific driver gene prediction and risk assessment through integrated network analysis of cancer omics profiles. Nucleic Acids Res. 43(3), 1332–1344 (2015)CrossRefGoogle Scholar
  4. 4.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: An efficient incremental algorithm for generating all maximal independent sets in hypergraphs of bounded dimension. Parallel Process. Lett. 10, 253–266 (2000)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Generating partial and multiple transversals of a hypergraph. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 588–599. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  6. 6.
    Damaschke, P.: Parameterizations of hitting set of bundles and inverse scope. J. Comb. Optim. 36(2012), 1–12 (2013)Google Scholar
  7. 7.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Eiter, T., Gottlob, G.: Hypergraph transversal computation and related problems in logic and AI. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 549–564. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  9. 9.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32(2), 514–537 (2003)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Eiter, T., Makino, K., Gottlob, G.: Computational aspects of monotone dualization: a brief survey. Discrete Appl. Math. 156(11), 2035–2049 (2008)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Gurvich, V., Khachiyan, L.: On generating the irredundant conjunctive and disjunctive normal forms of monotone boolean functions. Discrete Appl. Math. 96–97, 363–373 (1999)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hädicke, O., Klamt, S.: Computing complex metabolic intervention strategies using constrained minimal cut sets. Metab. Eng. 13(2), 204–213 (2011)CrossRefGoogle Scholar
  14. 14.
    Haus, U.-U., Klamt, S., Stephen, T.: Computing knock-out strategies in metabolic networks. J. Comput. Biol. 15(3), 259–268 (2008)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Jungreuthmayer, C., Nair, G., Klamt, S., Zanghellini, J.: Comparison and improvement of algorithms for computing minimal cut sets. BMC Bioinf. 14(1), 318 (2013)CrossRefGoogle Scholar
  16. 16.
    Khachiyan, L., Boros, E., Elbassioni, K., Gurvich, V.: An efficient implementation of a quasi-polynomial algorithm for generating hypergraph transversals and its application in joint generation. Discrete Appl. Math. 154(16), 2350–2372 (2006)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Khachiyan, L., Boros, E., Elbassioni, K., Gurvich, V.: A global parallel algorithm for the hypergraph transversal problem. Inf. Process. Lett. 101(4), 148–155 (2007)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Khachiyan, L., Boros, E., Gurvich, V., Elbassioni, K.: Computing many independent sets for hypergraphs in parallel. Parallel Process. Lett. 17(02), 141–152 (2007)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Murakami, K., Uno, T.: Efficient algorithms for dualizing large-scale hypergraphs. Discrete Appl. Math. 170, 83–94 (2014)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Sellis, T.K.: Multiple-query optimization. ACM Trans. Database Sys. 13(1), 23–52 (1988)CrossRefGoogle Scholar
  21. 21.
    Toda, T.: Hypergraph transversal computation with binary decision diagrams. In: Demetrescu, C., Marchetti-Spaccamela, A., Bonifaci, V. (eds.) SEA 2013. LNCS, vol. 7933, pp. 91–102. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ricardo Andrade
    • 2
    • 3
    • 4
    • 5
  • Etienne Birmelé
    • 1
  • Arnaud Mary
    • 2
    • 3
    • 4
    • 5
  • Thomas Picchetti
    • 1
  • Marie-France Sagot
    • 2
    • 3
    • 4
    • 5
  1. 1.MAP5, UMR CNRS 8145Université Paris DescartesParisFrance
  2. 2.Université de LyonLyonFrance
  3. 3.Université Lyon 1VilleurbanneFrance
  4. 4.CNRS, UMR5558Laboratoire de Biométrie et Biologie EvolutiveVilleurbanneFrance
  5. 5.INRIA Grenoble Rhône-Alpes - ERABLELyonFrance

Personalised recommendations