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 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.

[1]  Hans Kleine Büning,et al.  On subclasses of minimal unsatisfiable formulas , 2000, Discret. Appl. Math..

[2]  Hans Kleine Büning,et al.  The Complexity of Some Subclasses of Minimal Unsatis able Formulas , 2007, J. Satisf. Boolean Model. Comput..

[3]  V. Vinay,et al.  Branching rules for satisfiability , 1995, Journal of Automated Reasoning.

[4]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[5]  Igor Gashkov,et al.  New Optimal Constant Weight Codes , 2007, Electron. J. Comb..

[6]  Zsolt Tuza,et al.  One More Occurrence of Variables Makes Satisfiability Jump From Trivial to NP-Complete , 1993, SIAM J. Comput..

[7]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[8]  Marek Karpinski,et al.  Approximation Hardness and Satisfiability of Bounded Occurrence Instances of SAT , 2003, Electron. Colloquium Comput. Complex..

[9]  N. Alon,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2004 .

[10]  Christian Scheideler,et al.  A new algorithm approach to the general Lovász local lemma with applications to scheduling and satisfiability problems (extended abstract) , 2000, STOC '00.

[11]  Oliver Kullmann,et al.  The Combinatorics of Conflicts between Clauses , 2003, SAT.

[12]  Stefan Szeider,et al.  Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference , 2002, Theor. Comput. Sci..

[13]  Stefan Szeider,et al.  A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable , 2006, SIAM J. Discret. Math..

[14]  Linyuan Lu,et al.  Using Lovász Local Lemma in the Space of Random Injections , 2007, Electron. J. Comb..

[15]  Stefan Szeider,et al.  Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable , 2003, COCOON.

[16]  P. Erdos,et al.  The size Ramsey number , 1978 .

[17]  Paul Erdös,et al.  Lopsided Lovász Local Lemma and Latin transversals , 1991, Discret. Appl. Math..

[18]  Christos H. Papadimitriou,et al.  The complexity of facets resolved , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).