On the Exact Complexity of Evaluating Quantified k -CNF

Chris Calabro, Russell Impagliazzo, Ramamohan Paturi
2012 Algorithmica  
We relate the exponential complexities 2 s(k)n of k-sat and the exponential complexity 2 s(Π 2 3-sat)n of Π23-sat (evaluating formulas of the form ∀x∃yφ(x, y) where φ is a 3-CNF in x variables and y variables) and show that s(∞) (the limit of s(k) as k → ∞) is at most s(Π23-sat). Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for Π23-sat running in time 2 cn with c < 1. On the other hand, a nontrivial exponential-time algorithm for Π23-sat would
more » ... e a k-sat solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of Π23-sat have nontrivial algorithms, and provide strong evidence that the hardest cases of Π23-sat must have a mixture of clauses of two types: one universal literal and two existential literals, or only existential literals. Moreover, the hardest cases must have at least n − o(n) universal variables, and hence only o(n) existential variables. Our proofs involve the construction of efficient minimally unsatisfiable k-cnfs and the application of the Sparsification Lemma.
doi:10.1007/s00453-012-9648-0 fatcat:pqo7syfxsre4znksn3ds5ixwlu