Fixed-Parameter Tractable Reductions to SAT

ABSTRACT. Today's SAT solvers have an enormous importance and impact in many practical settings. They are used as efficient back-end to solve many NP-complete problems. However, many reasoning problems are located at the second level of the Polynomial Hierarchy or even higher, and hence polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. In certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions which exploit structural aspects of problem instances in terms of problem parameters. Recent research established a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not. We use this framework to analyze some problems that are related to Boolean satisfiability. We consider several natural parameterizations of these problems, and we identify for which of these an fpt-reduction to SAT is possible. The problems that we look at are related to minimizing an implicant of a DNF formula, minimizing a DNF formula, and satisfiability of quantified Boolean formulas.