Minimal Armstrong Databases for Cardinality Constraints

Hartmann et al. proved that calculating Armstrong instance for a collection of cardinality constraints is exactly exponential problem. In fact, they presented a collection of cardinality constraints based on some special graphs, whose minimal Armstrong instance is of exponential size. Motivated by that, graph based cardinality constraints are introduced in the present paper. That is, given a simple graph on the set of attributes, max cardinality constraints on edges end vertices, respectively are set. We take up the task to determine sizes of minimum Armstrong instances of graph based cardinality constraints for several graph classes, including bipartite graphs, complete multipartite graphs. We give exact results for several graph classes or give polynomial time algorithm to construct the minimum Armstrong table for other cases. We show that Armstrong tables of graph based cardinality constraints correspond to another graph defined on the maximal independent vertex sets of the constraint graph. The row graph of an Armstrong table is defined as a pair of rows form an edge if they contain identical entries in some column. It is shown that there exists a minimal Armstrong table for a collection of graph based cardinality constraints such that the line graph of its row graph is a spanning subgraph of the graph defined on the maximal independent sets of the constraint graph. This observation is used to find minimum Armstrong tables for bipartite constraint graphs. Feasible edge colourings of graphs introduced by Folkman and Fulkerson are used to construct minimum Armstrong tables for another class of constraint graphs.