A Fast Algorithm for SAT in Terms of Formula Length

In this paper, we prove that the general CNF satisfiability problem can be solved in $O^*(1.0646^L)$ time, where $L$ is the length of the input CNF-formula (i.e., the total number of literals in the formula), which improves the current bound $O^*(1.0652^L)$ given by Chen and Liu 12 years ago. Our algorithm is a standard branch-and-search algorithm analyzed by using the measure-and-conquer method. We avoid the bottleneck in Chen and Liu's algorithm by simplifying the branching operation for 4-degree variables and carefully analyzing the branching operation for 5-degree variables. To simplify case-analyses, we also introduce a general framework for analysis, which may be able to be used in other problems.