Enumeration Complexity of Conjunctive Queries with Functional Dependencies

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)

On the Enumeration of all Minimal Triangulations

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag

Unbalanced Triangle Detection and Enumeration Hardness for Unions of Conjunctive Queries

We study the enumeration of answers to Unions of Conjunctive Queries (UCQs) with optimal time guarantees. More precisely, we wish to identify the queries that can be solved with linear preprocessing time and constant delay. Despite the basic nature of this problem, it was shown only recently that UCQs can be solved within these time bounds if they admit free-connex union extensions, even if all individual CQs in the union are intractable with respect to the same complexity measure. Our goal is to understand whether there exist additional tractable UCQs, not covered by the currently known algorithms. As a first step, we show that some previously unclassified UCQs are hard using the classic 3SUM hypothesis, via a known reduction from 3SUM to triangle listing in graphs. As a second step, we identify a question about a variant of this graph task which is unavoidable if we want to classify all self-join free UCQs: is it possible to decide the existence of a triangle in a vertex-unbalanced tripartite graph in linear time? We prove that this task is equivalent in hardness to some family of UCQs. Finally, we show a dichotomy for unions of two self-join-free CQs if we assume the answer to this question is negative. Our conclusion is that, to reason about a class of enumeration problems defined by UCQs, it is enough to study the single decision problem of detecting triangles in unbalanced graphs. Without a breakthrough for triangle detection, we have no hope to find an efficient algorithm for additional unions of two self-join free CQs. On the other hand, if we will one day have such a triangle detection algorithm, we will immediately obtain an efficient algorithm for a family of UCQs that are currently not known to be tractable

Tuple-Independent Representations of Infinite Probabilistic Databases

Probabilistic databases (PDBs) are probability spaces over database instances. They provide a framework for handling uncertainty in databases, as occurs due to data integration, noisy data, data from unreliable sources or randomized processes. Most of the existing theory literature investigated finite, tuple-independent PDBs (TI-PDBs) where the occurrences of tuples are independent events. Only recently, Grohe and Lindner (PODS '19) introduced independence assumptions for PDBs beyond the finite domain assumption. In the finite, a major argument for discussing the theoretical properties of TI-PDBs is that they can be used to represent any finite PDB via views. This is no longer the case once the number of tuples is countably infinite. In this paper, we systematically study the representability of infinite PDBs in terms of TI-PDBs and the related block-independent disjoint PDBs. The central question is which infinite PDBs are representable as first-order views over tuple-independent PDBs. We give a necessary condition for the representability of PDBs and provide a sufficient criterion for representability in terms of the probability distribution of a PDB. With various examples, we explore the limits of our criteria. We show that conditioning on first order properties yields no additional power in terms of expressivity. Finally, we discuss the relation between purely logical and arithmetic reasons for (non-)representability

Database Repairing with Soft Functional Dependencies

A common interpretation of soft constraints penalizes the database for every violation of every constraint, where the penalty is the cost (weight) of the constraint. A computational challenge is that of finding an optimal subset: a collection of database tuples that minimizes the total penalty when each tuple has a cost of being excluded. When the constraints are strict (i.e., have an infinite cost), this subset is a "cardinality repair" of an inconsistent database; in soft interpretations, this subset corresponds to a "most probable world" of a probabilistic database, a "most likely intention" of a probabilistic unclean database, and so on. Within the class of functional dependencies, the complexity of finding a cardinality repair is thoroughly understood. Yet, very little is known about the complexity of finding an optimal subset for the more general soft semantics. This paper makes a significant progress in this direction. In addition to general insights about the hardness and approximability of the problem, we present algorithms for two special cases: a single functional dependency, and a bipartite matching. The latter is the problem of finding an optimal "almost matching" of a bipartite graph where a penalty is paid for every lost edge and every violation of monogamy

Tractable Orders for Direct Access to Ranked Answers of Conjunctive Queries

We study the question of when we can provide logarithmic-time direct access to the k-th answer to a Conjunctive Query (CQ) with a specified ordering over the answers, following a preprocessing step that constructs a data structure in time quasilinear in the size of the database. Specifically, we embark on the challenge of identifying the tractable answer orderings that allow for ranked direct access with such complexity guarantees. We begin with lexicographic orderings and give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orderings for every CQ without self-joins. We then continue to the more general orderings by the sum of attribute weights and show for it that ranked direct access is tractable only in trivial cases. Hence, to better understand the computational challenge at hand, we consider 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. We indeed achieve a quasilinear-time algorithm for a subset of the class of full CQs without self-joins, by adopting a solution of Frederickson and Johnson to the classic problem of selection over sorted matrices. We further prove that none of the other queries in this class admit such an algorithm.Comment: 17 page