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**1 - 4**of**4**### Fixed-Parameter Tractable Reductions to SAT 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 computational problems are located at the second level of the Polynomial Hierarchy or even higher, and hence po ..."

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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 computational 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.

### The complexity of SPP formula minimization

"... Circuit minimization is a useful procedure in the field of logic synthesis. Recently, it was proven that the minimization of (∨,∧,¬) formulae is hard for the second level of the polynomial hierarchy [BU08]. The complexity of minimizing more specialized formula models was left open, however. One mode ..."

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Circuit minimization is a useful procedure in the field of logic synthesis. Recently, it was proven that the minimization of (∨,∧,¬) formulae is hard for the second level of the polynomial hierarchy [BU08]. The complexity of minimizing more specialized formula models was left open, however. One model used in logic synthesis is a three-level model in which the third level is composed of parity gates, called SPPs. SPPs allow for small representations of Boolean functions and have efficient heuristics for minimization. However, little was known about the complexity of SPP minimization. Here, we show that SPP minimization is complete for the second level of the Polynomial Hierarchy under Turing reductions. ∗Supported by NSF CCF-0346991, CCF-0830787 and BSF 2004329.

### Acknowledgements

, 2008

"... Thanks to Chris Umans for help obtaining and writing up the results contained herein. ..."

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Thanks to Chris Umans for help obtaining and writing up the results contained herein.