Abstract
In this paper we present fixed-parameter algorithms for the problem Dual—given two hypergraphs, decide if one is the transversal hypergraph of the other—and related problems. In the first part, we consider the number of edges of the hypergraphs, the maximum degree of a vertex, and a vertex complementary degree as our parameters.
In the second part, we use an Apriori approach to obtain FPT results for generating all maximal independent sets of a hypergraph, all minimal transversals of a hypergraph, and all maximal frequent sets where parameters bound the intersections or unions of edges.
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References
Agrawal, R., Imielinski, T., Swami, A.N.: Mining association rules between sets of items in large databases. In: Proc. SIGMOD 1993, pp. 207–216 (1993)
Agrawal, R., Srikant, R.: Fast algorithms for mining association rules in large databases. In: Proc. VLDB 1994, pp. 487–499 (1994)
Boros, E., Elbassioni, K.M., Gurvich, V., Khachiyan, L.: An efficient incremental algorithm for generating all maximal independent sets in hypergraphs of bounded dimension. Parallel Processing Letters 10(4), 253–266 (2000)
Boros, E., Elbassioni, K.M., Gurvich, V., Khachiyan, L.: An inequality for polymatroid functions and its applications. Discrete Applied Mathematics 131(2), 255–281 (2003)
Boros, E., Elbassioni, K.M., Gurvich, V., Khachiyan, L.: Generating maximal independent sets for hypergraphs with bounded edge-intersections. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 488–498. Springer, Heidelberg (2004)
Boros, E., Gurvich, V., Hammer, P.L.: Dual subimplicants of positive Boolean functions. Optimization Methods and Software 10(2), 147–156 (1998)
Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 133–141. Springer, Heidelberg (2002)
Bioch, J.C., Ibaraki, T.: Complexity of identification and dualization of positive Boolean functions. Inf. Comput. 123(1), 50–63 (1995)
Damaschke, P.: Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theoret. Comput. Sci. 351(3), 337–350 (2006)
Damaschke, P.: The union of minimal hitting sets: Parameterized combinatorial bounds and counting. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 332–343. Springer, Heidelberg (2007)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
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.: Exact transversal hypergraphs and application to Boolean μ-functions. J. Symb. Comput. 17(3), 215–225 (1994)
Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)
Garcia-Molina, H., Barbará, D.: How to assign votes in a distributed system. J. ACM 32(4), 841–860 (1985)
Gunopulos, D., Khardon, R., Mannila, H., Saluja, S., Toivonen, H., Sharm, R.S.: Discovering all most specific sentences. ACM Trans. Database Syst. 28(2), 140–174 (2003)
Hagen, M.: On the fixed-parameter tractability of the equivalence test of monotone normal forms. Inf. Process. Lett. 103(4), 163–167 (2007)
Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)
Kavvadias, D.J., Stavropoulos, E.C.: Monotone Boolean dualization is in coNP[log2 n]. Inf. Process. Lett. 85(1), 1–6 (2003)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9(3), 558–565 (1980)
Makino, K.: Efficient dualization of O(logn)-term monotone disjunctive normal forms. Discrete Applied Mathematics 126(2–3), 305–312 (2003)
Mannila, H., Räihä, K.-J.: On the complexity of inferring functional dependencies. Discrete Applied Mathematics 40(2), 237–243 (1992)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237–252 (1975)
Sarkar, S., Sivarajan, K.N.: Hypergraph models for cellular mobile communication systems. IEEE Transactions on Vehicular Technology 47(2), 460–471 (1998)
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Elbassioni, K., Hagen, M., Rauf, I. (2008). Some Fixed-Parameter Tractable Classes of Hypergraph Duality and Related Problems. In: Grohe, M., Niedermeier, R. (eds) Parameterized and Exact Computation. IWPEC 2008. Lecture Notes in Computer Science, vol 5018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79723-4_10
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DOI: https://doi.org/10.1007/978-3-540-79723-4_10
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