Degrees of acyclicity for hypergraphs and relational database schemes

Ronald Fagin
1983 Journal of the ACM  
Database schemes (winch, intuitively, are collecuons of table skeletons) can be wewed as hypergraphs (A hypergraph Is a generalization of an ordinary undirected graph, such that an edge need not contain exactly two nodes, but can instead contain an arbitrary nonzero number of nodes.) A class of "acychc" database schemes was recently introduced. A number of basic desirable propemes of database schemes have been shown to be equivalent to acyclicity This shows the naturalness of the concept.
more » ... r, unlike the situation for ordinary, undirected graphs, there are several natural, noneqmvalent notions of acyclicity for hypergraphs (and hence for database schemes). Various desirable properties of database schemes are constdered and it is shown that they fall into several equivalence classes, each completely characterized by the degree of acycliclty of the scheme The results are also of interest from a purely graph-theoretic viewpomt. The original notion of aeyclicity has the countermtmtive property that a subhypergraph of an acychc hypergraph can be cyclic. This strange behavior does not occur for the new degrees of acyelicity that are considered. SUPPLIER PART PROJECT PARTICOST I !1 515 FIGURE 1 FIGURE 2 problems that are NP-complete in general, but which have polynomial-time algorithms under the assumption of a-acyclicity. There are other, even nicer properties of database schemes that are too strong to be obeyed by all a-acyclic database schemes. We study some such properties and characterize graph-theoretically those database schemes which enjoy these properties. Once again, the properties fall into equivalence classes, which correspond to natural "degrees of acyclicity" for hypergraphs. For, unlike the situation for ordinary, undirected graphs, there are a number ofinequivalent, natural definitions of acyclicity for hypergraphs. It is appropriate to speak of "degrees of acydicity," rather than simply "types of acychcity," since it turns out that there is a linear ordering of the strengths of the types of acyclicity we consider; the weakest (least restrictive) is the previously studied notion of a-acyclicity. Our new degrees of acychcity remedy a mathematically unnatural property of the earlier notion of a-acyclicity; namely, it is possible for a hypergraph to be a-acyclic but have an t~-cyclic subhypergraph. (A subhypergraph contains a subset of the edges of the original hypergraph.) This strange phenomenon does not occur for our new degrees of acyclicity. Each of the degrees of hypergraph acyclicity that we consider is a generalization of the concept of acyclicity for ordinary undirected graphs; that is, an undirected graph is acyclic in the usual sense if and only if it is "acyclic," when viewed as a hypergraph, for any of our notions of "acyclic." There is an analogy between degrees of acyclicity for database schemes and normal forms [ 15, 20] for relation schemata (a relation scheme along with its set of dependencies [21] ). For, there is a hierarchy of normal forms for relation schemata, each
doi:10.1145/2402.322390 fatcat:emmduwyygvdj5ekrlrtj7wtwpu