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Register a new userGreedy Strategies and Larger Islands of Tractability for Conjunctive Queries and Constraint Satisfaction Problems
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Added by
Francesco Scarcello
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Structural decomposition methods have been developed for identifying tractable classes of instances of fundamental problems in databases, such as conjunctive queries and query containment, of the constraint satisfaction problem in artificial intelligence, or more generally of the homomorphism problem over relational structures. Most structural decomposition methods can be characterized through hypergraph games that are variations of the Robber and Cops graph game that characterizes the notion of treewidth. In particular, decomposition trees somehow correspond to monotone winning strategies, where the escape space of the robber on the hypergraph is shrunk monotonically by the cops. In fact, unlike the treewidth case, there are hypergraphs where monotonic strategies do not exist, while the robber can be captured by means of more complex non-monotonic strategies. However, these powerful strategies do not correspond in general to valid decompositions. The paper provides a general way to exploit the power of non-monotonic strategies, by allowing a disciplined form of non-monotonicity, characteristic of cops playing in a greedy way. It is shown that deciding the existence of a (non-monotone) greedy winning strategy (and compute one, if any) is tractable. Moreover, despite their non-monotonicity, such strategies always induce valid decomposition trees, which can be computed efficiently based on them. As a consequence, greedy strategies allow us to define new islands of tractability for the considered problems properly including all previously known classes of tractable instances.
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Evaluating conjunctive queries and solving constraint satisfaction problems are fundamental problems in database theory and artificial intelligence, respectively. These problems are NP-hard, so that several research efforts have been made in the literature for identifying tractable classes, known as islands of tractability, as well as for devising clever heuristics for solving efficiently real-world instances. Many heuristic approaches are based on enforcing on the given instance a property called local consistency, where (in database terms) each tuple in every query atom matches at least one tuple in every other query atom. Interestingly, it turns out that, for many well-known classes of queries, such as for the acyclic queries, enforcing local consistency is even sufficient to solve the given instance correctly. However, the precise power of such a procedure was unclear, but for some very restricted cases. The paper provides full answers to the long-standing questions about the precise power of algorithms based on enforcing local consistency. The classes of instances where enforcing local consistency turns out to be a correct query-answering procedure are however not efficiently recognizable. In fact, the paper finally focuses on certain subclasses defined in terms of the novel notion of greedy tree projections. These latter classes are shown to be efficiently recognizable and strictly larger than most islands of tractability known so far, both in the general case of tree projections and for specific structural decomposition methods.
Marx (STOC~2010, J.~ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class $Phi$ of Boolean CQs of bounded submodular width, the model-checking problem for $Phi$ on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marxs result to non-Boolean queries as follows: For every class $Phi$ of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marxs splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently.
A dominant cost for query evaluation in modern massively distributed systems is the number of communication rounds. For this reason, there is a growing interest in single-round multiway join algorithms where data is first reshuffled over many servers and then evaluated in a parallel but communication-free way. The reshuffling itself is specified as a distribution policy. We introduce a correctness condition, called parallel-correctness, for the evaluation of queries w.r.t. a distribution policy. We study the complexity of parallel-correctness for conjunctive queries as well as transferability of parallel-correctness between queries. We also investigate the complexity of transferability for certain families of distribution policies, including, for instance, the Hypercube distribution.
The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called islands of tractability. A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language, whereas a constraint language is conservative if it contains all unary relations. This paper addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages isnt tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose we utilize the notion of a emph{strong backdoor} of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated moves the instance to an island of tractability, i.e., to a tractable class of instances. In this paper, we consider strong backdoors into emph{scattered classes}, consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Our main result is an algorithm that, given a CSP instance with $n$ variables, finds in time $f(k)n^{O(1)}$ a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size $k$ or correctly decides that there isnt such a backdoor.
We investigate the clustering transition undergone by an exemplary random constraint satisfaction problem, the bicoloring of $k$-uniform random hypergraphs, when its solutions are weighted non-uniformly, with a soft interaction between variables belonging to distinct hyperedges. We show that the threshold $alpha_{rm d}(k)$ for the transition can be further increased with respect to a restricted interaction within the hyperedges, and perform an asymptotic expansion of $alpha_{rm d}(k)$ in the large $k$ limit. We find that $alpha_{rm d}(k) = frac{2^{k-1}}{k}(ln k + ln ln k + gamma_{rm d} + o(1))$, where the constant $gamma_{rm d}$ is strictly larger than for the uniform measure over solutions.
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