Hard satisfiable formulas for DPLL-type algorithms

Satisfiability is one of the most popular NP-complete problems. There are two main types of algorithms for solving SAT, namely local search (for references see, for example, [3]) and DPLL-type (this type was first described in the work [5] of Davis and Putnam and [4] of Davis, Logemann and Loveland). A lot of effort has been invested in proving ”less-that-2N ” upper bounds for such algorithms. In this paper we concentrate on proving exponential lower bounds and consider two DPLL-type algorithms: GUC (Generalized Unit Clause heuristic; introduced in [2]) and Randomized GUC. DPLL-type algorithms were historically the first “less-than-2N” algorithms for SAT. They receive as input a formula F in CNF with variables x1, . . . , xN . After that, a DPLL-type algorithm simplifies the input according to a certain set of transformation rules. If the answer now is obvious (the simplified formula is either empty or contains a pair of contradicting unit clauses), the algorithm returns an answer. In the opposite case, it chooses a literal l in the formula according to a certain heuristic. Then it constructs two formulas, one corresponding to l := true and the other to l := false, and recursively calls itself for these two formulas (note that since we deal with the running time of the algorithm, the order in which it calls itself for these two formulas does matter). If any of the calls returns the answer “Satisfiable”, the algorithm also returns this answer. Otherwise, it returns “Unsatisfiable”. Therefore, such algorithms differ from each other by two procedures: one for simplifying a formula, and the other for choosing the next literal.