Abstract
Hypergraph Dualization (also called as hitting set enumeration) is the problem of enumerating all minimal transversals of a hypergraph \({\mathcal {H}}\), i.e., all minimal inclusion-wise hyperedges (i.e., sets of vertices) that intersect every hyperedge in \({\mathcal {H}}\). Dualization is at the core of many important Artificial Intelligence (AI) problems. As a contribution to a better understanding of Dualization complexity, this paper introduces a tight upper bound to the number of minimal transversals that can be computed in polynomial time. In addition, the paper presents an interesting exploitation of the upper bound to the number of minimal transversals. In particular, the problem dealt with is characterizing the complexity of the data mining problem called \(\mathtt {IFM}_{\mathtt {I}}\) (Inverse Frequent itemset Mining with Infrequency constraints), that is the problem of finding a transaction database whose frequent and infrequent itemsets satisfy a number of frequency/infrequency patterns given in input.
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- 1.
A preliminary version of this paper is a companion manuscript for [16].
- 2.
Other hypergraph definitions require a hyperedge not to be empty.
- 3.
We assume that \(\log _{\bar{n}} \widetilde{m}^d_{3^+}\) is equal to \(-\infty \) if \(\widetilde{m}^d_{3^+}=0\) or it is rounded up to a fixed number of decimal places otherwise.
References
Agrawal, R., Imieliński, T., Swami, A.: Mining association rules between sets of items in large databases. In: Proceedings of the 1993 ACM SIGMOD International Conference on Management of Data, SIGMOD 1993, pp. 207–216. ACM, New York (1993). https://doi.org/10.1145/170035.170072
Berge, C.: Graphs and Hypergraphs. North-Holland Pub. Co., Amsterdam (1973)
Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On maximal frequent and minimal infrequent sets in binary matrices. Ann. Math. Artif. Intell. 39, 211–221 (2003). https://doi.org/10.1023/A:1024605820527
Calders, T.: Itemset frequency satisfiability: complexity and axiomatization. Theoret. Comput. Sci. 394(1–2), 84–111 (2008). https://doi.org/10.1016/j.tcs.2007.11.003
Damaschke, P.: Parameterized algorithms for double hypergraph dualization with rank limitation and maximum minimal vertex cover. Discret. Optim. 8(1), 18–24 (2011). https://doi.org/10.1016/j.disopt.2010.02.006
Desaulniers, G., Desrosiers, J., Solomon, M.M.: Column Generation. Springer, New York (2005). https://doi.org/10.1007/b135457
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9
Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995). https://doi.org/10.1137/S0097539793250299
Eiter, T., Makino, K.: Generating all abductive explanations for queries on propositional horn theories. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 197–211. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45220-1_18
Elbassioni, K.M., Rauf, I., Ray, S.: Enumerating minimal transversals of geometric hypergraphs. In: Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, 10–12 August (2011)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X
Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996). https://doi.org/10.1006/jagm.1996.0062
Gottlob, G., Malizia, E.: Achieving new upper bounds for the hypergraph duality problem through logic. SIAM J. Comput. 47(2), 456–492 (2018). https://doi.org/10.1137/15M1027267
Gottlob, G.: Deciding monotone duality and identifying frequent itemsets in quadratic logspace. In: Hull, R., Fan, W. (eds.) PODS, pp. 25–36. ACM (2013). https://doi.org/10.1145/2463664.2463673
Gunopulos, D., Khardon, R., Mannila, H., Toivonen, H.: Data mining, hypergraph transversals, and machine learning. In: Mendelzon, A.O., Özsoyoglu, Z.M. (eds.) PODS 1997, pp. 209–216. ACM Press (1997). https://doi.org/10.1145/263661.263684
Guzzo, A., Moccia, L., Saccà, D., Serra, E.: Solving inverse frequent itemset mining with infrequency constraints via large-scale linear programs. ACM Trans. Knowl. Discov. Data 7(4), 18:1–18:39 (2013). https://doi.org/10.1145/2541268.2541271
Kavvadias, D., Papadimitriou, C.H., Sideri, M.: On horn envelopes and hypergraph transversals. In: Ng, K.W., Raghavan, P., Balasubramanian, N.V., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 399–405. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57568-5_271
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). https://doi.org/10.1016/j.tcs.2007.03.005
Liu, G., Li, J., Wong, L.: A new concise representation of frequent itemsets using generators and a positive border. Knowl. Inf. Syst. 17(1), 35–56 (2008). https://doi.org/10.1007/s10115-007-0111-5
Mielikainen, T.: On inverse frequent set mining. In: Proceedings of 2nd Workshop on Privacy Preserving Data Mining, PPDM 2003, pp. 18–23. IEEE Computer Society, Washington, DC (2003)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Boston (1994)
Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987). https://doi.org/10.1016/0004-3702(87)90062-2
Saccà, D., Serra, E.: On line appendix to: number of minimal hypergraph transversals and complexity of IFM with infrequency: high in theory, but often not so much in practice! Version of 12 September 2019. http://sacca.deis.unical.it/#view=object&format=object&id=1490/gid=160
Saccà, D., Serra, E., Guzzo, A.: Count constraints and the inverse OLAP problem: definition, complexity and a step toward aggregate data exchange. In: Lukasiewicz, T., Sali, A. (eds.) FoIKS 2012. LNCS, vol. 7153, pp. 352–369. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28472-4_20
Saccá, D., Serra, E., Rullo, A.: Extending inverse frequent itemsets miningto generate realistic datasets: complexity, accuracy and emerging applications. Data Min. Knowl. Discov. 33, 1736–1774 (2019). https://doi.org/10.1007/s10618-019-00643-1
Vardi, M.Y.: Lost in math? Commun. ACM 62(3), 7 (2019). https://doi.org/10.1145/3306448
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Saccà, D., Serra, E. (2019). Number of Minimal Hypergraph Transversals and Complexity of IFM with Infrequency: High in Theory, but Often Not so Much in Practice!. In: Alviano, M., Greco, G., Scarcello, F. (eds) AI*IA 2019 – Advances in Artificial Intelligence. AI*IA 2019. Lecture Notes in Computer Science(), vol 11946. Springer, Cham. https://doi.org/10.1007/978-3-030-35166-3_14
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