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We study the enumeration complexity of Unions of Conjunctive Queries (UCQs). We aim to identify the UCQs that are tractable in the sense that the answer tuples can be enumerated with a linear preprocessing phase and a constant delay between every successive tuples. It has been established that, in the absence of self joins and under conventional complexity assumptions, the CQs that admit such an evaluation are precisely the free-connex ones. A union of tractable CQs is always tractable. We generalize the notion of freeconnexity from CQs to UCQs, thus showing that some unions containing intractable CQs are, in fact, tractable. Interestingly, some unions consisting of only intractable CQs are tractable too. The question of a finding a full characterization of the tractability of UCQs remains open. Nevertheless, we prove that for several classes of queries, free-connexity fully captures the tractable UCQs.According to the dichotomy of Bagan et al. [2], the enumeration problem for Q 2 is in DelayC lin , while Q 1 is intractable.

We study the question of when we can provide direct access to the k -th answer to a Conjunctive Query (CQ) according to a specified order over the answers in time logarithmic in the size of the database, following a preprocessing step that constructs a data structure in time quasilinear in database size. Specifically, we embark on the challenge of identifying the tractable answer orderings , that is, those orders that allow for such complexity guarantees. To better understand the computational challenge at hand, we also investigate the more modest task of providing access to only a single answer (i.e., finding the answer at a given position), a task that we refer to as the selection problem , and ask when it can be performed in quasilinear time. We also explore the question of when selection is indeed easier than ranked direct access. We begin with lexicographic orders . For each of the two problems, we give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orders for every CQ without self-joins. We then continue to the more general orders by the sum of attribute weights and establish the corresponding decidable characterizations, for each of the two problems, of the tractable CQs without self-joins. Finally, we explore the question of when the satisfaction of Functional Dependencies (FDs) can be utilized for tractability, and establish the corresponding generalizations of our characterizations for every set of unary FDs.

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We study the complexity of enumerating the answers of Conjunctive Queries (CQs) in the presence of Functional Dependencies (FDs). Our focus is on the ability to list output tuples with a constant delay in between, following a linear-time preprocessing. A known dichotomy classifies the acyclic self-join-free CQs into those that admit such enumeration, and those that do not. However, this classification no longer holds in the common case where the database exhibits dependencies among attributes. That is, some queries that are classified as hard are in fact tractable if dependencies are accounted for. We establish a generalization of the dichotomy to accommodate FDs; hence, our classification determines which combination of a CQ and a set of FDs admits constant-delay enumeration with a linear-time preprocessing.In addition, we generalize a hardness result for cyclic CQs to accommodate a common type of FDs. Further conclusions of our development include a dichotomy for enumeration with linear delay, and a dichotomy for CQs with disequalities. Finally, we show that all our results apply to the known class of "cardinality dependencies" that generalize FDs (e.g., by stating an upper bound on the number of genres per movies, or friends per person). XX:3reduction fails, however, when dependencies are imposed on the data, as the constructed database instance does not necessarily satisfy the underlying dependencies.As it turns out, however, the structure of the FD-extended query Q + allows us to extend this reduction to our setting. By carefully expanding the reduced instance such that on the one hand, the dependencies hold and on the other hand, the reduction can still be performed within linear time, we establish a dichotomy. That is, we show that the tractability of enumerating the answers of a self-join-free query Q in the presence of FDs is exactly characterized by the structure of Q + : Given an FD-acyclic query Q, we can enumerate the answers to Q within the class DelayC lin iff Q is FD-free-connex.The resulting extended dichotomy, as well as the original one, brings insight to the case of acyclic queries. Concerning unrestricted CQs, providing even a first solution of a query in linear time is impossible in general. This is due to the fact that the parameterized complexity of answering boolean CQs, taking the query size as the parameter, is W[1]-hard [13]. This does not imply, however, that there are no cyclic queries with the corresponding enumeration problems in DelayC lin . The fact that no such queries exist requires an additional proof, which was presented by Brault- Baron [6]. This result holds under a generalization of the triangle finding problem, which is considered not to be solvable within linear time [16]. As before, this proof does no longer apply in the presence of FDs. Moreover, it is possible for Q to be cyclic and Q + acyclic. In fact, Q + may even be free-connex, and therefore tractable in DelayC lin . We show that, under the same assumptions used by Brault-Baron [6], the evaluation problem for a s...