One hierarchy spawns another: graph deconstructions and the complexity classification of conjunctive queries

Hubie Chen, M. Müller
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引用次数: 15

Abstract

We study the problem of conjunctive query evaluation relative to a class of queries; this problem is formulated here as the relational homomorphism problem relative to a class of structures A, wherein each instance must be a pair of structures such that the first structure is an element of A. We present a comprehensive complexity classification of these problems, which strongly links graph-theoretic properties of A to the complexity of the corresponding homomorphism problem. In particular, we define a binary relation on graph classes and completely describe the resulting hierarchy given by this relation. This binary relation is defined in terms of a notion which we call graph deconstruction and which is a variant of the well-known notion of tree decomposition. We then use this hierarchy of graph classes to infer a complexity hierarchy of homomorphism problems which is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized version of quantifier-free reductions. In doing so, we obtain a significantly refined complexity classification of homomorphism problems, as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications. We then present and develop the theory of Ehrenfeucht-Fraïssé-style pebble games which solve the homomorphism problems where the cores of the structures in A have bounded tree depth. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank.
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一个层次结构衍生出另一个层次结构:图解构和联合查询的复杂性分类
研究了一类查询的联合查询求值问题;本文将此问题表述为一类结构a的关系同态问题,其中每个实例必须是一对结构,使得第一个结构是a的一个元素。我们提出了这些问题的综合复杂性分类,它将a的图论性质与相应同态问题的复杂性紧密联系起来。特别地,我们在图类上定义了一个二元关系,并完整地描述了这种关系所给出的层次结构。这种二元关系是根据我们称之为图解构的概念来定义的,它是众所周知的树分解概念的一个变体。然后,我们使用图类的层次结构来推断同态问题的复杂性层次结构,该层次结构是全面的,直到计算上非常弱的约简概念,即无量词约简的参数化版本。在此过程中,我们获得了同态问题的非常精细的复杂性分类,以及对现有复杂性分类的统一、模块化和概念清晰的处理。然后,我们提出并发展了Ehrenfeucht-Fraïssé-style卵石对策理论,该理论解决了A中结构核具有有界树深度的同态问题。最后,利用该框架对量词秩有界的模型检验存在句的复杂度进行了分类。
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