An Algorithm for Dualization in Products of Lattices and Its Applications

Let \cL=\cL_1×⋅s×\cL_n be the product of n lattices, each of which has a bounded width. Given a subset \cA\subseteq\cL, we show that the problem of extending a given partial list of maximal independent elements of \cA in \cL can be solved in quasi-polynomial time. This result implies, in particular, that the problem of generating all minimal infrequent elements for a database with semi-lattice attributes, and the problem of generating all maximal boxes that contain at most a specified number of points from a given n-dimensional point set, can both be solved in incremental quasi-polynomial time

Extending the Balas-Yu Bounds on the Number of Maximal Independent Sets in Graphs to Hypergraphs and Lattices

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most # + 1, where # is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors in the product of n lattices L1 Ln , where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into B. We sho