On Acyclic Conjunctive Queries and Constant Delay Enumeration

• Published in: Computer Science Logic Vol. 4646; pp. 208 - 222
• Main Authors: Bagan, Guillaume, Durand, Arnaud, Grandjean, Etienne
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2007; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Computer science; Mathematical theory of computation; Programming & scripting languages: general
• ISBN: 3540749144; 9783540749141
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-540-74915-8_18

We study the enumeration complexity of the natural extension of acyclic conjunctive queries with disequalities. In this language, a number of NP-complete problems can be expressed. We first improve a previous result of Papadimitriou and Yannakakis by proving that such queries can be computed in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c.|\cal M|.|\varphi(\cal M)|$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\cal M$\end{document} is the structure, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi(\cal M)$\end{document} is the result set of the query and c is a simple exponential in the size of the formula ϕ. A consequence of our method is that, in the general case, tuples of such queries can be enumerated with a linear delay between two tuples. We then introduce a large subclass of acyclic formulas called CCQ ≠  and prove that the tuples of a CCQ ≠  query can be enumerated with a linear time precomputation and a constant delay between consecutive solutions. Moreover, under the hypothesis that the multiplication of two n×n boolean matrices cannot be done in time O(n2), this leads to the following dichotomy for acyclic queries: either such a query is in CCQ ≠  or it cannot be enumerated with linear precomputation and constant delay. Furthermore we prove that testing whether an acyclic formula is in CCQ ≠  can be performed in polynomial time. Finally, the notion of free-connex treewidth of a structure is defined. We show that for each query of free-connex treewidth bounded by some constant k, enumeration of results can be done with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(|{\mathcal M}|^{k+1})$\end{document} precomputation steps and constant delay.