Research Article

Compression with wildcards: Abstract simplicial complexes

Published in: Quaestiones Mathematicae
Volume 46 , issue 6 , 2023 , pages: 1151–1173

DOI: 10.2989/16073606.2022.2071181

Author(s): Marcel Wild Stellenbosch University, SA

Keywords: 68P05 , 68R07 , 68W10 , 05E45 , Abstract simplicial complex , face-numbers , antitone Boolean function , exclusive sum of products , binary decision diagram , compressed enumeration , wildcards , reliability polynomial , partitionability conjecture

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Abstract

Despite the more handy terminology of abstract simplicial complexes , in its core this article is about antitone Boolean functions. Given the maximal faces (=facets) of , our main algorithm, called Facets-To-Faces, outputs 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.

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