Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
A Computing Procedure for Quantification Theory
Journal of the ACM (JACM)
A Machine-Oriented Logic Based on the Resolution Principle
Journal of the ACM (JACM)
Satisfiability, Branch-Width and Tseitin Tautologies
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Fourier meets möbius: fast subset convolution
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
Algorithms for propositional model counting
Journal of Discrete Algorithms
Trimmed Moebius Inversion and Graphs of Bounded Degree
Theory of Computing Systems - Special Title: Symposium on Theoretical Aspects of Computer Science; Guest Editors: Susanne Albers, Pascal Weil
Combinatorial Optimization on Graphs of Bounded Treewidth
The Computer Journal
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We consider unsatisfiable Boolean formulas in conjunctive normal form. It is known that unsatisfiability can be shown by a regular resolution proof in time polynomial in the number of variables n, and exponential in the tree-width w. It is also known that satisfiability for bounded tree-width can actually be decided in time linear in the length of the formula and exponential in the tree-width w. We investigate the complexities of resolution proofs and arbitrary proofs in more detail depending on the number of variables n and the tree-width w. We present two very traditional algorithms, one based on resolution and the other based on truth tables. Maybe surprisingly, we point out that the two algorithms turn out to be basically the same algorithm with different interpretations. Similar results holds for a bound w′ on the tree-width of the incidence graph for a somewhat extended notion of a resolution proof. The length of any proper resolution proof might be quadratic in n, but if we allow to introduce abbreviations for frequently occurring subclauses, then the proof length and running time are again linear in n.