Unsatisfiable Linear CNF Formulas Are Large and Complex

Author Dominik Scheder

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Dominik Scheder

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Dominik Scheder. Unsatisfiable Linear CNF Formulas Are Large and Complex. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 621-632, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


We call a CNF formula {\em 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 $4k^24^k$ clauses, and on the other hand, any linear $k$-CNF formula with at most $\frac{4^k}{8e^2k^2}$ 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 $2^{2^{\frac{k}{2}-1}}$. 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 {\em strict} resolution tree, i.e. every clause of $F$ is used only once in the resolution tree, then we need at least $a^{a^{\iddots^a}}$ clauses, where $a \approx 2$ and the height of this tower is roughly $k$.
  • Extremal combinatorics
  • proof complexity
  • probabilistic method


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