International Workshop on Graph Structures for Knowledge Representation and Reasoning

Graph Structures for Knowledge Representation and Reasoning pp 72-88 | Cite as

Combinatorial Results on Directed Hypergraphs for the SAT Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9501)


Directed hypergraphs have already been shown to unveil several combinatorial inspired results for the SAT problem. In this paper we approach the SAT problem by searching a transversal of the directed hypergraphs associated to its instance. We introduce some particular clause orderings and study their influence on the backtrack process, exhibiting a new subclass of CNF for which SAT is polynomial. Based on unit resolution and a novel dichotomous search, a new DPLL-like algorithm and a renaming-based combinatorial approach are proposed. We then investigate the study of weak transversals in this setting and reveal a new degree of a CNF formula unsatisfiability and a structural result about unsatisfiable formulae.




  1. 1.
    Ausiello, G.: Directed hypergraphs: data structures and applications. In: Dauchet, M., Nivat, M. (eds.) CAAP 1988. LNCS, vol. 299, pp. 295–303. Springer, Heidelberg (1988)Google Scholar
  2. 2.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7, 201–215 (1960)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. In: Proceedings of 34th ACM Symposium on Theory of Computing, Montreal, Quebec, Canada, 19–21 May 2002Google Scholar
  5. 5.
    Fiduccia, C.M., Mattheyses, R.M.: A linear time heuristic for improving network partitions. In: Proceedings of ACM/IEEE Design Automation Conference, pp. 175–181 (1982)Google Scholar
  6. 6.
    Gallo, G., Gentile, C., Pretolani, D., Rago, G.: Max Horn sat and the minimum cut problem in directed hypergraphs. Math. Program. 80, 213–237 (1998)MathSciNetMATHGoogle Scholar
  7. 7.
    Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discrete Appl. Math. 42, 177–201 (1993)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  9. 9.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified np-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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.) Algorithms and Computation. LNCS, vol. 762, pp. 399–405. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  11. 11.
    Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49, 291–307 (1970)CrossRefMATHGoogle Scholar
  12. 12.
    Kullmann, O.: An application of matroid theory to the sat problem. Technical report, ECCC TR00-018 (2000)Google Scholar
  13. 13.
    Lewis, H.: Renaming a set of clauses as a horn set. J. Assoc. Comput. Mach. 25, 134–135 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Porschen, S., Speckenmeyer, E., Randerath, B.: On linear CNF formulas. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 212–225. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Torres, A.F., Araoz, J.D.: Combinatorial models for searching in knowledge bases. Mathematicas Acta Cient. Venez. 39, 387–394 (1988)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University Al. I. CuzaIasiRomania
  2. 2.LIRMMUniversity MontpellierMontpellierFrance

Personalised recommendations