On Average Case Complexity of SAT for Symmetric Distribution
Johann A. Makowsky, Abraham Sharell · Journal of Logic and Computation · 1995
Abstract We investigate in this paper ‘natural’ distributions for the satisfiability problem (SAT) of prepositional logic, using concepts previously introduced by to study the average-case complexity of NP-complete problems. Gurevich showed that a problem with a flat distribution is not DistNP complete (for deterministic reductions), unless DEXPTIme ≠ NEXPTlme. We express the known results concerning fixed size and fixed density distributions for CNF in the framework of average-case complexity and show that all these distributions are flat. We introduce the family of symmetric distributions, which generalizes those mentioned before, and show that bounded symmetric distributions on ordered tuples of clauses (CNFTupIes) and on k-CNF (sets of k-literal-clauses), are flat. This eliminates all these distributions as candidates for ‘provably hard’ (i.e. DistNP complete) distributions for SAT, if one considers only deterministic reductions. Given the (presumed) naturalness and generality of these distributions, this result supports evidence that (at least polynomial-time, no-error) randomized reductions are appropriate in average-case complexity. We also observe, that there are non-flat distributions for which SAT is polynomial on the average, but that this is due to the particular choice of the size functions. Finally, Chváal and Szemerédi have shown that for certain fixed size distributions (which are also flat) resolution is exponential for almost all instances. We use this to show that every resolution algorithm will need at least exp(nα) (for any 0 ≤ α ≤ 1) time on the average. In other words, resolution-based algorithms will not establish that SAT, with these distributions, is in AverP.