Counting solutions to conjunctive queries

Abstract

Counting the number of answers to conjunctive queries is an intractable problem, formally #P-hard, even over classes of acyclic queries. However, Durand and Mengel have recently introduced the notion of quantified star size that, combined with hypertree decompositions, identifies islands of tractability for the problem. They also wonder whether such a notion precisely characterizes those classes for which the counting problem is tractable. We show that this is the case only for bounded-arity simple queries, where relation symbols cannot be shared by different query atoms.Indeed, we give a negative answer to the question in the general case, by exhibiting a more powerful structural method based on the novel concept of #-generalized hypertree decomposition. On classes of queries with bounded #-generalized hypertree width, counting answers is shown to be feasible in polynomial time, after a fixed-parameter polynomial-time preprocessing that only depends on the query structure. A weaker variant (but still more general than the technique based on the quantified starsize) is also proposed, for which tractability is established without any exponential dependency on the query size.Based on #-generalized hypertree decompositions, a "hybrid" decomposition method is eventually conceived, where structural properties of the query are exploited in combination with properties of the given database, such as keys or other (weaker) dependencies among attributes that limit the allowed combinations of values. Intuitively, such features may induce different structural properties that are not identified by the "worst-possible database" perspective of purely structural methods.

Answering Conjunctive Queries under Updates

“…The result has recently been extended to all conjunctive queries over a fixed schema by Chen and Mengel [11]. Structural properties that make the counting problem for CQs tractable in the case where the schema is part of the input have been identified in [16,18]. Join Evaluation.…”

Computing the Shapley value in allocation problems: approximations and bounds, with an application to the Italian VQR research assessment program

“…There are however some promising recent advances that identify islands of tractability for the allocation problems where at most one good is allocated to each agent: it has been recently shown that those instances where the treewidth of the agents' interaction-graph is bounded by some constant (i.e., have a low degree of cyclicity) can be solved in polynomial-time (Greco et al, 2015). The result is based on recent advances on counting solutions of conjunctive queries with existential variables (Greco & Scarcello, 2014a). Unfortunately, if the structure is quite cyclic this technique cannot be applied to large instances, because its computational complexity has an exponential dependency on the treewidth.…”

Fast Counting in Machine Learning Applications

“…1, the query Count((X 1 = 3)∧(X 2 = 2)∧(X 3 = 1)) would return 2, as there are 2 instances matching the query condition. We note that the above formulation of counting is a special and simple case of the general counting problem in conjunctive queries, known from database theory [9] (we provide more details in Section 5).…”

“…Support for counting queries is a primary component in any database management system. In such systems, the query mechanism must support conjunctive queries over multiple tables, and with a variety of possible query predicates [9]. Moreover, the queries are typically executed over tables that cannot be fully materialized in the main memory.…”