A Better Analysis For PPSZ For 3-SAT

2026-07-12

AI Summary

This work improves the upper bound analysis of the randomized algorithm PPSZ for 3-SAT. Without modifying the original algorithm or the lifting theorem from Unique-3-SAT to General 3-SAT, we reconstruct the complexity analysis framework by introducing structured coordinate unification and replacing the traditional recombination step with an explicit linear programming dual certificate. This approach uniquely integrates linear programming duality theory with exact rational interval arithmetic, significantly enhancing analytical precision. While preserving the original estimation structure, our method yields tighter worst-case running time bounds: $O^*(1.306969598^n)$ for Unique-3-SAT and $O^*(1.307031578^n)$ for General 3-SAT, with the latter constituting the current best-known result.

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.

Problem

• 3-SAT
• PPSZ algorithm
• running-time bound
• Unique-3-SAT
• randomized algorithm

Innovation

• PPSZ algorithm
• linear programming dual certificate
• 3-SAT
• running-time analysis
• exact rational interval computation