How Many Conflicts Does It Need to Be Unsatisfiable?

A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. Clearly, a CNF formula has to have conflicts in order to be unsatisfiable—in fact, there have to be many conflicts, and it is the goal of this paper to quantify how many. An unsatisfiable k-CNF has at least 2 clauses; a lower bound of 2 for the number of conflicts follows easily. We improve on this trivial bound by showing that an unsatisfiable k-CNF formula requires Ω(2.32) conflicts. On the other hand there exist unsatisfiable k-CNF formulas with O( 4 k log k k ) conflicts. This improves the simple bound O(4) arising from the unsatisfiable k-CNF formula with the minimum number of clauses.