We i n v estigate the problem of rewriting queries with aggregate operators using views that may o r m a y not contain aggregate operators. A rewriting of a query is a second query that uses view predicates such that evaluating rst the views and then the rewriting yields the same result as evaluating the original query. In this sense, the original query and the rewriting are equivalent modulo the view de nitions. The queries and views we consider correspond to unnested SQL queries, possibly with union, that employ the operators min, max, count, and sum.Our approach is based on syntactic characterizations of the equivalence of aggregate queries. One contribution of this paper are characterizations of the equivalence of disjunctive aggregate queries, which generalize our previous results for the conjunctive case.For each operator , w e i n troduce several types of queries using views as candidates for rewritings. We unfold such a candidate by replacing each occurrence of a view predicate with its de nition, thus obtaining a regular aggregate query. The candidates have a di erent, usually more complex operator than . We prove that unfolding the candidate, however, results in a regular aggregate query that is equivalent to the candidate modulo the view de nitions. This property justi es considering these types of queries as natural candidates for rewritings. In this way, we reduce the problem of whether there exist rewritings of a particular type to a problem involving equivalence.We distinguish between partial rewritings that contain at least one view predicate and complete rewritings that contain only view predicates. In contrast to previous work on this topic, we not only give su cient, but also necessary conditions for a rewriting to exist. More precisely, w e show for each t ype of candidate that the existence of both, partial and complete rewritings is decidable, and we provide upper and lower complexity bounds. IntroductionRewriting queries using views is a fundamental problem in databases, which has attracted considerable attention. View usability techniques have applications in a number of areas. In query optimization, the execution of a query can be accelerated if results from previous queries can be used to compute answers YL87, CR94, CKPS95 . In designing information systems over which a h uge number of a priori known queries are posed periodically, it can be bene cial to store such intermediate results beforehand that are useful for as many queries as possible LFS97, RSS96 . Integrating heterogeneous information sources is another problem which may be reduced to the view usability problem LSK95 .While the focus of this work was for a long time on queries without aggregation, interest in aggregate queries has been motivated recently by the surge of data warehousing and decision support applications, where queries of this kind typically occur. Optimization based on the reuse of previously computed results is particularly promising for aggregate queries, since often huge numbers of data items are p...

This paper investigates a graph enumeration problem, called the maximal P-subgraphs problem, where P is a hereditary or connected-hereditary graph property. Formally, given a graph G, the maximal P-subgraphs problem is to generate all maximal induced subgraphs of G that satisfy P. This problem differs from the well-known node-deletion problem, studied by Yannakakis and Lewis [J. Lewis, On the complexity of the maximum subgraph J. Lewis, M. Yannakakis, The node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 (2) (1980) 219-230]. In the maximal P-subgraphs problem, the goal is to produce all (locally) maximal subgraphs of a graph that have property P, whereas in the node-deletion problem, the goal is to find a single (globally) maximum size subgraph with property P. Algorithms are presented that reduce the maximal P-subgraphs problem to an input-restricted version of this problem. These algorithms imply that when attempting to efficiently solve the maximal P-subgraphs problem for a specific P, it is sufficient to solve the restricted case. The main contributions of this paper are characterizations of when the maximal P-subgraphs problem is in a complexity class C (e.g., polynomial delay, total polynomial time).

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The problem of enumerating (i.e., generating) all maximal cliques in a graph has received extensive treatment, due to the plethora of applications in various areas such as data mining, bioinformatics, network analysis and community detection. However, requiring the enumerated subgraphs to be full cliques is too restrictive in common real-life scenarios where "almost cliques" are equally useful. Hence, the notion of a k-plex, a clique relaxation that allows every node to be "missing" k neighbors, has been introduced. But this seemingly minor relaxation casts existing algorithms for clique enumeration inapplicable, for inherent reasons. This paper presents the first provably efficient algorithms, both for enumerating the maximal k-plexes and for enumerating the maximal connected k-plexes. Our algorithms run in polynomial delay for a constant k and incremental FPT delay when k is a parameter. The importance of such algorithms is in the areas mentioned above, as well as in new applications. Extensive experimentation over both real and synthetic datasets shows the efficiency of our algorithms, and their scalability with respect to graph size, density and choice of k, as well as their clear superiority over the stateof-the-art.

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