Graph minors. V. Excluding a planar graph
Journal of Combinatorial Theory Series B
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
On the hardness of approximate reasoning
Artificial Intelligence
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Satisfiability, Branch-Width and Tseitin Tautologies
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Constraint Processing
Algorithms and Complexity Results for #SAT and Bayesian Inference
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Backdoor Sets for DLL Subsolvers
Journal of Automated Reasoning
Solving #SAT using vertex covers
Acta Informatica
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms
Algorithmica - Parameterized and Exact Algorithms
Improved algorithms for feedback vertex set problems
Journal of Computer and System Sciences
AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 1
Backdoors to typical case complexity
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
Algorithms for propositional model counting
Journal of Discrete Algorithms
The Multivariate Algorithmic Revolution and Beyond
Parameterized Complexity
The Multivariate Algorithmic Revolution and Beyond
Upper and lower bounds for weak backdoor set detection
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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Backdoor sets contain certain key variables of a CNF formula F that make it easy to solve the formula. More specifically, a weak backdoor set of F is a set X of variables such that there exits a truth assignment τ to X that reduces F to a satisfiable formula F[τ] that belongs to a polynomial-time decidable base class $\mathcal C$. A strong backdoor set is a set X of variables such that for all assignments τ to X, the reduced formula F[τ] belongs to $\mathcal C$. We study the problem of finding backdoor sets of size at most k with respect to the base class of CNF formulas with acyclic incidence graphs, taking k as the parameter. We show that 1 the detection of weak backdoor sets is W[2]-hard in general but fixed-parameter tractable for r-CNF formulas, for any fixed r≥3, and 2 the detection of strong backdoor sets is fixed-parameter approximable. Result 1 is the the first positive one for a base class that does not have a characterization with obstructions of bounded size. Result 2 is the first positive one for a base class for which strong backdoor sets are more powerful than deletion backdoor sets. Not only SAT, but also #SAT can be solved in polynomial time for CNF formulas with acyclic incidence graphs. Hence Result 2 establishes a new structural parameter that makes #SAT fixed-parameter tractable and that is incomparable with known parameters such as treewidth and clique-width. We obtain the algorithms by a combination of an algorithmic version of the Erdős-Pósa Theorem, Courcelle's model checking for monadic second order logic, and new combinatorial results on how disjoint cycles can interact with the backdoor set.