# Compression with wildcards: Abstract simplicial complexes

## TL;DR

This paper introduces a new algorithm called Facets-To-Faces for efficiently compressing and representing abstract simplicial complexes, with applications in various computational fields.

## Contribution

The paper presents a novel algorithm for compressing simplicial complexes using wildcards, improving efficiency over existing methods, and introduces a new way to compute face numbers from facets.

## Key findings

- Facets-To-Faces outperforms Mathematica's BooleanConvert and Python BDDs in compression.
- The algorithms can be parallelized for enhanced performance.
- Applications include reliability analysis, combinatorial topology, and frequent set mining.

## 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 in high-level Mathematica code, compares favorably to both the Mathematica command BooleanConvert, and to the BDD's 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.