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Register a new userSub-exponential complexity of regular linear CNF formulas
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Added by
Bernd Schuh
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Bernd R. Schuh
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The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e. XSAT can be determined in sub-exponential time. Here I show that this class is just a subset of a larger class of LCNF formulas which display this very kind of complexity. To this end I introduce the property of disjointedness of LCNF formulas, measured, for a single clause C, by the number of clauses which have no variable in common with C. If for a given LCNF formula F all clauses have the same disjointedness d we call F d-disjointed and denote the class of such formulas by dLCNF. XLCNF formulas correspond to the special cased=0. One main result of the paper is that the class of all monotone l-regular LCNF formulas which are d-disjointed, with d smaller than some upper bound D, is of sub-exponential complexity. This result can be generalized to show that all monotone, l-regular LCNF formulas F which have a bounded mean disjointedness, are of sub-exponential XSAT-complexity, as well.
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Open questions with respect to the computational complexity of linear CNF formulas in connection with regularity and uniformity are addressed. In particular it is proven that any l-regular monotone CNF formula is XSAT-unsatisfiable if its number of clauses m is not a multiple of l. For exact linear formulas one finds surprisingly that l-regularity implies k-uniformity, with m = 1 + k(l-1)) and allowed k-values obey k(k-1) = 0 (mod l). Then the computational complexity of the class of monotone exact linear and l-regular CNF formulas with respect to XSAT can be determined: XSAT-satisfiability is either trivial, if m is not a multiple of l, or it can be decided in sub-exponential time, namely O(exp(n^^1/2)). Sub-exponential time behaviour for the wider class of regular and uniform linear CNF formulas can be shown for certain subclasses.
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