Structural Tractability of Counting of Solutions to Conjunctive Queries

Abstract

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 n O(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.

The logic of counting query answers

“…Define contractpA, Sq to be the graph on vertices S obtained by starting from GrSs and adding an edge between any two vertices that appear together in an D-component of pA, Sq. 3 Example 4.1: Let φpu 1 , u 2 , u 3 , u 4 q be the pp-formula…”

“…Previous work showed that, for bounded arity Φ, countpΦq is fixed-parameter tractable if Φ satisfies the tractability condition; interreducible with the parameterized clique problem if Φ satisfies the contraction condition but not the tractability condition; and as hard as the parameterized counting clique problem otherwise (see [4] for a precise statement). 3 Note that in previous articles [4], [18] this graph was defined in terms of the core of the given formula.…”

Constant Delay Enumeration for Conjunctive Queries

“…For instance it is known that the boolean case is tractable for CQ having bounded width for various notions of width. In order to capture also non boolean queries the notion of quantified-star size was extended for various notions of width [13]. Based on this definition for hypertreewidth, the result reads as follows: Theorem 8.…”

Decomposing Quantified Conjunctive (or Disjunctive) Formulas

“…Another direction is to study the complexity of counting answers to formulas; in this problem, one is given a first-order formula and a finite structure, and the goal is to count the number of assignments, to the free variables of the formula, that satisfy the formula on the structure. There have been recent results on this direction in the context of primitive positive formulas [28,23] and existential positive formulas [24].…”