CNF satisfiability test by counting and polynomial average time
SIAM Journal on Computing
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
On the hardness of approximate reasoning
Artificial Intelligence
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
Fixed-parameter complexity in AI and nonmonotonic reasoning
Artificial Intelligence
On the Clique-Width of Perfect Graph Classes
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Algorithms and Complexity Results for #SAT and Bayesian Inference
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
The Parameterized Complexity of Counting Problems
SIAM Journal on Computing
Hidden Structure in Unsatisfiable Random 3-SAT: An Empirical Study
ICTAI '04 Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence
Graph-Modeled Data Clustering: Exact Algorithms for Clique Generation
Theory of Computing Systems
Approximating clique-width and branch-width
Journal of Combinatorial Theory Series B
Backdoor Sets for DLL Subsolvers
Journal of Automated Reasoning
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
The backdoor key: a path to understanding problem hardness
AAAI'04 Proceedings of the 19th national conference on Artifical intelligence
Backbones and backdoors in satisfiability
AAAI'05 Proceedings of the 20th national conference on Artificial intelligence - Volume 3
Parameterized Complexity
Learning to assign degrees of belief in relational domains
Machine Learning
Matched formulas and backdoor sets
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
Backdoor sets of quantified boolean formulas
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
Algorithms for propositional model counting
LPAR'07 Proceedings of the 14th international conference on Logic for programming, artificial intelligence and reasoning
Complexity and algorithms for well-structured k-SAT instances
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
A new bound for an NP-hard subclass of 3-SAT using backdoors
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
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We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clustering-width. Our algorithm runs in uniform polynomial time on formulas with bounded clustering-width. It is known that the number of models of formulas with bounded clique-width, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clustering-width and the other parameters mentioned are incomparable: there are formulas with bounded clustering-width and arbitrarily large clique-width, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clustering-width and bounded clique-width, treewidth, and branchwidth.