Compression with wildcards: Abstract simplicial complexes

Abstract

Despite the more handy terminology of abstract simplicial complexes SC, in its core this article is about antitone Boolean functions. Given  the maximal faces (=facets) of SC, our main algorithm, called Facets-To-Faces, outputs SC in a compressed format. The degree of  compression of Facets-To-Faces, which is programmed using the high-level programming language Mathematica, compares favorably to  both the hardwired Mathematica command BooleanConvert, and to the hardwired binary decision diagrams provided by Python. A novel  way to calculate the face-numbers from the facets is also presented. Both algorithms can be parallelized and are applicable (e.g.) to  reliability analysis, combinatorial topology, and frequent-set-mining.