The Parameterized Complexity of Relational Database Queries and an Improved Characterization of W[1]

It is well known that for a fixed relational database query φ in m free variables, it can be determined in time polynomial in the size n of the database whether there exists an m-tuple x that belongs to the relation defined by the query. For the best known algorithms, however, the exponent of the polynomial is proportional to the size of the query. We study the data complexity of this problem parameterized by the size k = |φ| of the query, and answer a question recently raised by Yannakakis [Yan95]. Our main results show: (1) the general problem is complete for the parametric complexity class AW [∗], and (2) when restricted to monotone queries, the problem is complete for the fundamental parametric complexity class W [1]. The practical significance of these results is that unless the parameterized complexity hierarchy collapses, there are unlikely to be algorithms that solve this problem (even under the restriction to monotone queries) in time f(k)n where f is an arbitrary function of k and c is a constant independent of k. An important consequence of the proof of (2) is a significantly improved characterization of the parameterized complexity class W [1]. Previous results by Downey and Fellows characterize W [1] in terms of the kWeighted Circuit Satisfiability problem, for families of circuits that satisfy: (1) the depth of the circuits is bounded by a constant c, (2) on any input-output path there is at most one gate having unbounded fanin (termed a large gate), with all other gates having fan-in bounded by c (that is, small gates). We show that the definition can be broadened by allowing circuits of depth bounded by an arbitrary function f(k). If we denote this parameterized complexity class W ∗[1], then our corollary ⋆ Research supported by the New Zealand Marsden Fund for Basic Science ⋆⋆ Research supported by the National Science and Engineering Research Council of Canada. is: W ∗[1] = W [1]. As a consequence, the k-Weighted Satisfiability problem for Boolean expressions that are products of k-sums is complete for W [1]. Part of our argument establishes a general normalization lemma for W ∗[t] for all t.