A Better Analysis For PPSZ For 3-SAT

Abstract

We revisit Scheder's analysis of the original PPSZ algorithm. Keeping his regular and irregular estimates unchanged, we express them in common structural coordinates and replace only their final recombination by an explicit linear-programming dual certificate. The old and new running-time bounds are \[ \begin{array}{c|cc} & \text{Unique-$3$-SAT} & \text{general $3$-SAT} \\ \hline \text{Scheder's analysis} & O^*(1.306972377^n) & O^*(1.307031594^n) \\ \text{this work} & O^*(1.306969598^n) & O^*(1.307031578^n). \end{array} \] In both rows, the general-case bound is obtained by applying the same existing Scheder--Steinberger unique-to-general lifting theorem to the corresponding Unique-$3$-SAT analysis. To the best of our knowledge, $O^*(1.307031578^n)$ is the best currently known worst-case randomized running-time bound for general $3$-SAT. Neither PPSZ nor the lifting theorem is modified. The numerical inequalities are certified by exact rational interval computation.