How many conflicts does it need to be unsatisfiable?

Authors:
Dominik Scheder;Philipp Zumstein
Affiliations:
Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland;Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland
Venue:
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
Year:
2008

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Abstract

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 2k clauses; a lower bound of 2k for the number of conflicts follows easily. We improve on this trivial bound by showing that an unsatisfiable k-CNF formula requires Ω(2.32k) conflicts. On the other hand there exist unsatisfiable k-CNF formulas with O(4k log3 k/k) conflicts. This improves the simple bound O(4k) arising from the unsatisfiable k-CNF formula with the minimum number of clauses.