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- W2962941898 abstract "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 $k$." @default.
- W2962941898 created "2019-07-30" @default.
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- W2962941898 date "2010-01-01" @default.
- W2962941898 modified "2023-09-23" @default.
- W2962941898 title "Unsatisfiable Linear CNF Formulas Are Large and Complex." @default.
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- W2962941898 doi "https://doi.org/10.4230/lipics.stacs.2010.2490" @default.
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