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.
R. Andrade—Financially supported by CNPq – Brazil.
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References
Angel, E., Bampis, E., Gourvès, L.: On the minimum hitting set of bundles problem. Theoret. Comput. Sci. 410(45), 4534–4542 (2009)
Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North-Holland, Amsterdam (1989)
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)
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)
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)
Damaschke, P.: Parameterizations of hitting set of bundles and inverse scope. J. Comb. Optim. 36(2012), 1–12 (2013)
Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)
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)
Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32(2), 514–537 (2003)
Eiter, T., Makino, K., Gottlob, G.: Computational aspects of monotone dualization: a brief survey. Discrete Appl. Math. 156(11), 2035–2049 (2008)
Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)
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)
Hädicke, O., Klamt, S.: Computing complex metabolic intervention strategies using constrained minimal cut sets. Metab. Eng. 13(2), 204–213 (2011)
Haus, U.-U., Klamt, S., Stephen, T.: Computing knock-out strategies in metabolic networks. J. Comput. Biol. 15(3), 259–268 (2008)
Jungreuthmayer, C., Nair, G., Klamt, S., Zanghellini, J.: Comparison and improvement of algorithms for computing minimal cut sets. BMC Bioinf. 14(1), 318 (2013)
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)
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)
Khachiyan, L., Boros, E., Gurvich, V., Elbassioni, K.: Computing many independent sets for hypergraphs in parallel. Parallel Process. Lett. 17(02), 141–152 (2007)
Murakami, K., Uno, T.: Efficient algorithms for dualizing large-scale hypergraphs. Discrete Appl. Math. 170, 83–94 (2014)
Sellis, T.K.: Multiple-query optimization. ACM Trans. Database Sys. 13(1), 23–52 (1988)
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)
Acknowledgements
We would like to thank the reviewers for their remarks which helped us to make the paper clearer, and particularly for pointing out the pre-existing notion of hitting sets of bundles.
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Andrade, R., Birmelé, E., Mary, A., Picchetti, T., Sagot, MF. (2015). Incremental Complexity of a Bi-objective Hypergraph Transversal Problem. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_16
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