Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k2 4k clauses, and on the other hand, any linear k-CNF formula with at most 4k 8e2k2 clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 22 k 2 −1 . This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least aa . .. a clauses, where a ≈ 2 and the height of this tower is roughly k.

• Type: Conference
• Year: 2010
• Where: STACS
• Authors: Dominik Scheder
• Keywords: CNF Formula Linear, Linear K-cnf Formulas, STACS 2010, Theoretical Computer Science, Unsatisfiable Linear