3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General

The PPSZ algorithm by Paturi, Pudl'ak, Saks, and Zane [1998] is the fastest known algorithm for Unique k-SAT, where the input formula does not have more than one satisfying assignment. For k>=5 the same bounds hold for general k-SAT. We show that this is also the case for k=3,4, using a slightly modified PPSZ algorithm. We do the analysis by defining a cost for satisfiable CNF formulas, which we prove to decrease in each PPSZ step by a certain amount. This improves our previous best bounds with Moser and Scheder [2011] for 3-SAT to O(1.308^n) and for 4-SAT to O(1.469^n).

💡 Research Summary

The paper revisits the classic PPSZ algorithm—originally introduced by Paturi, Pudlák, Saks, and Zane in 1998—as the fastest known randomized algorithm for Unique k‑SAT, where the input formula has at most one satisfying assignment. For k ≥ 5 it was already known that the same exponential time bound applies to general (non‑unique) k‑SAT. The authors show that this equivalence also holds for the notoriously difficult cases k = 3 and k = 4, by employing a slightly modified version of PPSZ and a novel “cost” analysis.

The modified algorithm replaces the traditional s‑bounded resolution preprocessing step with a weaker but more convenient notion called s‑implication. A literal is s‑implied if it can be forced to true by a sub‑formula consisting of at most s clauses. During the execution of PPSZ the algorithm repeatedly applies s‑implication until no further literals are implied, then proceeds to assign the next variable (chosen uniformly at random) either by the forced value (if a unit clause or s‑implication exists) or by a random guess. This change does not weaken the algorithm’s power because any unit clause that would appear in the original analysis also appears as an s‑implication in the new setting.

The central technical contribution is the definition of a cost function c(F) for any satisfiable CNF formula F. The cost is bounded by S·n, where n is the number of variables and S is a constant derived from the known bound Sₖ on the probability that a frozen variable (one that takes the same value in all satisfying assignments) is guessed incorrectly. For frozen variables the guessing probability is at most Sₖ + εₖ(s), where εₖ(s) → 0 as s grows. Non‑frozen variables have higher guessing probabilities, but their presence also raises the chance that the current partial assignment remains satisfiable after a random guess. The authors prove that in each PPSZ step the expected decrease in the cost is at least 1. Consequently, starting from an initial cost ≤ S·n, after at most S·n steps the cost reaches zero, which corresponds to having found a satisfying assignment. The probability of success is therefore at least 2^{‑c(F)} ≥ 2^{‑S·n}, yielding a running time of O(2^{S·n}) for the overall algorithm.

For k = 3 the constant S₃ evaluates to 2 ln 2 − 1 ≈ 0.3862, giving a running time bound of O(1.30704ⁿ). For k = 4, S₄ ≈ 0.5548, leading to O(1.46899ⁿ). These improve upon the previous best bounds for general 3‑SAT (≈ 1.32065ⁿ) and 4‑SAT (≈ 1.46928ⁿ) established by Moser and Scheder (2011). The improvement is modest in absolute terms but significant because it matches the bound known for Unique k‑SAT, demonstrating that the “uniqueness” assumption does not give any asymptotic advantage for these values of k.

The analysis also simplifies earlier proofs that relied on intricate “critical clause trees.” By focusing on the cost function and the s‑implication process, the authors avoid the need for complex combinatorial arguments, making the proof more transparent and potentially easier to adapt to other settings. Moreover, the same framework extends naturally to larger k, reproducing the known results for k ≥ 5 without additional work.

In summary, the paper provides a clean, unified analysis of PPSZ that bridges the gap between Unique k‑SAT and general k‑SAT for k = 3 and 4, delivering the best known exponential‑time bounds for these problems and offering a more accessible proof technique that may influence future research on exact algorithms for SAT and related constraint satisfaction problems.