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Structure and Complexity of Bag Consistency
ACM SIGMOD Record ( IF 1.1 ) Pub Date : 2022-06-01 , DOI: 10.1145/3542700.3542719
Albert Atserias, Phokion G. Kolaitis

Since the early days of relational databases, it was realized that acyclic hypergraphs give rise to database schemas with desirable structural and algorithmic properties. In a bynow classical paper, Beeri, Fagin, Maier, and Yannakakis established several different equivalent characterizations of acyclicity; in particular, they showed that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for relations over that schema holds, which means that every collection of pairwise consistent relations over the schema is globally consistent. Even though real-life databases consist of bags (multisets), there has not been a study of the interplay between local consistency and global consistency for bags. We embark on such a study here and we first show that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for bags over that schema holds. After this, we explore algorithmic aspects of global consistency for bags by analyzing the computational complexity of the global consistency problem for bags: given a collection of bags, are these bags globally consistent? We show that this problem is in NP, even when the schema is part of the input. We then establish the following dichotomy theorem for fixed schemas: if the schema is acyclic, then the global consistency problem for bags is solvable in polynomial time, while if the schema is cyclic, then the global consistency problem for bags is NP-complete. The latter result contrasts sharply with the state of affairs for relations, where, for each fixed schema, the global consistency problem for relations is solvable in polynomial time.



中文翻译:

包一致性的结构和复杂性

从关系数据库的早期开始,人们就意识到非循环超图会产生具有理想结构和算法属性的数据库模式。在一篇 bynow 经典论文中,Beeri、Fagin、Maier 和 Yannakakis 建立了几种不同的非循环性等价表征;特别是,他们表明模式的属性集形成一个非循环超图当且仅当该模式上的关系的局部到全局一致性属性成立,这意味着模式上的每个成对一致关系的集合都是全局的持续的。尽管现实生活中的数据库由袋子(多组)组成,但还没有研究过袋子的局部一致性和全局一致性之间的相互作用。我们在这里开始这样的研究,我们首先表明,当且仅当该模式上的包的局部到全局一致性属性成立时,模式的属性集形成一个非循环超图。在此之后,我们通过分析包的全局一致性问题的计算复杂性来探索包的全局一致性的算法方面:给定一个包的集合,这些包是全局一致的吗?我们证明了这个问题存在于 NP 中,即使模式是输入的一部分。然后我们为固定模式建立了以下二分定理:如果模式是非循环的,那么袋子的全局一致性问题在多项式时间内是可解决的,而如果模式是循环的,那么袋子的全局一致性问题是 NP 完全的。后一个结果与关系的状况形成鲜明对比,其中,

更新日期:2022-06-02
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