Common method for solving satisfiability problems which lie in P
Schaefer's dichotomy theorem is proved by dividing CSPs into two types: those that can be reduced to one of a few specific problems in P, and the other to which SAT can be reduced (and so are NP-complete). Specifically, every CSP of the former type is either trivial (always satisfied by the constant 0 or the constant 1 assignment), can be reduced to 2SAT, can be reduced to HORN-SAT, or can be reduced to XOR-SAT. These are the only algorithms you need to solve these CSPs. There is no one single algorithm – there is a finite list of algorithms.