An algorithm for the class of pure implicational formulas
Discrete Applied Mathematics - Special issue on the satisfiability problem and Boolean functions
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Complexity of the Falsifiability Problem for Pure Implicational Formulas
MFCS '95 Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Journal of Computer and System Sciences
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Consider a forest of k trees and n nodes together with a (partial) function σ mapping leaves of the trees to non-root nodes of other trees. Define the shadow of a leaf l to be the subtree rooted at σ(l). The shadow problem asks whether there is a set S of leaves exactly one from each tree such that none of these leaves lies in the shadow of another leaf in S. This graph theoretical problem as shown in Franco et al. (Discrete Appl. Math. 96 (1999) 89) is equivalent to the falsifiability problem for pure implicational Boolean formulas over n variables with k occurences of the constant false as introduced in: Heusch J. Wiedermann, P. Hajek (Edso), Proceedings of the Twentieth International Symposium on Mathematical Foundations of Computer Science (MFCS'95), Prague, Czech Republic, Lecture Notes in Computer Science, Vol. 969, Springer, Berlin, 1995, pp. 221-226, where its NP-completeness is shown for arbitrary values of k and a time bound of O(nk) for fixed k was obtained. In Franco et al. (1999) this bound is improved to O(n2kk) showing the problem's fixed parameter tractability (Congr. Numer. 87 (1992) 161). In this paper the bound O(n33k) is achieved by dynamic programming techniques thus significantly improving the fixed parameter part.