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Structure and Complexity of Bag Consistency
arXiv - CS - Databases Pub Date : 2020-12-22 , DOI: arxiv-2012.12126
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 by-now 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:

制袋一致性的结构和复杂性

自从关系数据库成立以来,人们就认识到非循环超图会产生具有所需结构和算法属性的数据库模式。在现在的古典论文中,比里(Beeri),法金(Fagin),迈尔(Maier)和扬纳卡基斯(Yannakakis)建立了几种不同的无环性等效描述;特别是,他们表明,当且仅当该模式之间的关系的局部到全局一致性属性成立时,该模式的属性集才形成一个非循环超图,这意味着该模式上成对一致关系的每个集合都是全局的一致的。即使现实生活中的数据库由袋子(多个集合)组成,也没有研究袋子的局部一致性和全局一致性之间的相互作用。我们在这里着手进行这样的研究,我们首先证明,当且仅当该模式上的包的局部到全局一致性属性成立时,该模式的属性集才形成一个非循环超图。此后,我们通过分析袋子的全局一致性问题的计算复杂性来探索袋子的全局一致性的算法方面:给定袋子的集合,这些袋子是否全局一致?我们证明,即使模式是输入的一部分,这个问题也存在于NP中。然后,我们为固定模式建立以下二分定理:如果模式是非循环的,则袋的全局一致性问题可以在多项式时间内解决;而如果模式是循环的,则袋的全局一致性问题是NP-完全的。后一种结果与关系的事态形成鲜明对比,在这种情况下,
更新日期:2020-12-23
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