Abstract
A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is Π 2P P-complete, answering a question posed by Kleine Büning and Zhao.
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References
R. Aharoni and N. Linial, Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas, J. Combin. Theory Ser. A 43 (1986) 196–204.
R. Diestel, Graph Theory, 2nd ed., Graduate Texts in Mathematics, Vol. 173 (Springer, New York, 2000).
R.G. Downey and M.R. Fellows, Parameterized Complexity (Springer, 1999).
T. Eiter and G. Gottlob, Complexity results for some eigenvector problems, International Journal of Computer Mathematics 76(1) (2000) 59–74.
H. Fleischner, O. Kullmann and S. Szeider, Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference, Theoretical Computer Science 289(1) (2002) 503–516.
J. Franco and A. Van Gelder, A perspective on certain polynomial time solvable classes of satisfiability, Discrete Applied Mathematics 125 (2003) 177–214.
M.R. Garey and D.R. Johnson, Computers and Intractability (Freeman, New York, 1979).
P. Hell, Graph packings, in: Proceedings for the 6th International Conference on Graph Theory, ed. I. Rusu, Electronic Notes in Discrete Mathematics, Vol. 5 (2000).
H. Kleine Büning and T. Lettman, Propositional Logic: Deduction and Algorithms (Cambridge Univ. Press, Cambridge, 1999).
H. Kleine Büning and X. Zhao, Satisfiable formulas closed under replacement, in: Proceedings for the Workshop on Theory and Applications of Satisfiability, eds. H. Kautz and B. Selman, Electronic Notes in Discrete Mathematics, Vol. 9 (2001).
O. Kullmann, Investigations on autark assignments, Discrete Applied Mathematics 107(1–3) (2000) 99–137.
O. Kullmann, Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets, Discrete Applied Mathematics 130(2) (2003) 209–249.
L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Mathematics, Vol. 29 (North-Holland, Amsterdam, 1986).
C.H. Papadimitriou, Computational Complexity (Addison-Wesley, 1994).
M. Schaefer, Completeness in the polynomial-time hierarchy, Technical Report TR01-009, School of CTI, DePaul University, Chicago, IL (2001).
L.J. Stockmeyer, The polynomial-time hierarchy, Theoretical Computer Science 3(1) (1976) 1–22.
S. Szeider, Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable, in: Proceedings of the 9th International Computing and Combinatorics Conference (COCOON’03), eds. T. Warnow and B. Zhu, Lecture Notes in Computer Science, Vol. 2697 (2003) pp. 548–558.
C.A. Tovey, A simplified NP-complete satisfiability problem, Discrete Applied Mathematics 8(1) (1984) 85–89.
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Szeider, S. Generalizations of matched CNF formulas. Ann Math Artif Intell 43, 223–238 (2005). https://doi.org/10.1007/s10472-005-0432-6
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DOI: https://doi.org/10.1007/s10472-005-0432-6