The Efficiency of Resolution and Davis--Putnam Procedures

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiable formula of F finds a resolution proof of F , and the runtime of our algorithm is nontrivial as a function of the size of the shortest resolution proof of F . Next we investigate a class of backtrack search algorithms, commonly known as Davis-Putnam procedures and provide the first average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is 2Θ(n(n/m) 1/(k−2)) with probability 1− o(1). Finally, we give new lower bounds on res(F), the size of the smallest resolution refutation of F , for a class of formulas representing the pigeonhole principle, and for randomly generated formulas. For random formulas, Chvátal and Szemerédi had shown that random 3-CNF formulas with a linear number of clauses require exponential size resolution proofs ∗Preliminary versions of these results appeared in the 37th IEEE Symposium on Foundations of Computer Science [BP96] and the 30th ACM Symposium of Theory of Computing [BKPS98]. †Research supported by NSF grant CCR-9303017. ‡ Research supported by NSF grant CCR-9457782 and US-Israel BSF Grant 95-00238. § Research supported by NSF grant CCR-9700239. This work was done while the author was on sabbatical at University of Washington.