A comparison of structural CSP decomposition methods
Artificial Intelligence
When is the evaluation of conjunctive queries tractable?
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Query evaluation via tree-decompositions
Journal of the ACM (JACM)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
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The complexity of counting homomorphisms seen from the other side
Theoretical Computer Science
Constraint solving via fractional edge covers
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The complexity of homomorphism and constraint satisfaction problems seen from the other side
Journal of the ACM (JACM)
Hypertree width and related hypergraph invariants
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A unified theory of structural tractability for constraint satisfaction problems
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Generalized hypertree decompositions: NP-hardness and tractable variants
Journal of the ACM (JACM)
Approximating fractional hypertree width
ACM Transactions on Algorithms (TALG)
Structural tractability of enumerating CSP solutions
CP'10 Proceedings of the 16th international conference on Principles and practice of constraint programming
Journal of Computer and System Sciences
Decomposing Quantified Conjunctive (or Disjunctive) Formulas
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
On acyclic conjunctive queries and constant delay enumeration
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
The complexity of weighted counting for acyclic conjunctive queries
Journal of Computer and System Sciences
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In this paper we explore the problem of counting solutions to conjunctive queries. We consider a parameter called the quantified star size of a formula ϕ which measures how the free variables are spread in ϕ. We show that for conjunctive queries that admit nice decomposition properties (such as being of bounded treewidth or generalized hypertree width) bounded quantified star size exactly characterizes the classes of queries for which counting the number of solutions is tractable. This also allows us to fully characterize the conjunctive queries for which counting the solutions is tractable in the case of bounded arity. To illustrate the applicability of our results, we also show that computing the quantified star size of a formula is possible in time nO(k) for queries of generalized hypertree width k. Furthermore, quantified star size is even fixed parameter tractable parameterized by some other width measures, while it is W[1]-hard for generalized hypertree width and thus unlikely to be fixed parameter tractable. We finally show how to compute an approximation of quantified star size in polynomial time where the approximation ratio depends on the width of the input.