How to partition or count an abstract simplicial complex, given its facets

TL;DR

This paper presents an efficient method for partitioning an abstract simplicial complex into compact representations using wildcards, improving over traditional techniques like inclusion-exclusion and BDDs for applications in optimization and algebra.

Contribution

It introduces a novel partitioning approach that encodes complexes with wildcards, enabling faster face-number calculations and better optimization performance.

Findings

Partitioning is more efficient than face-number calculation alone.

Method outperforms inclusion-exclusion and binary decision diagrams.

Applicable to optimization, algebra, and data mining tasks.

Abstract

Given are the facets of an abstract (finite) simplicial complex SC. We show how to partition SC into few pieces, each one compactly encoded by the use of wildcards. Such a representation is useful for the optimization of a target function SC -> Z, as well as in combinatorial commutative algebra and Frequent Set Mining. Merely calculating the face-numbers of SC can be done faster than partitioning SC. Our method compares favorably to inclusion-exclusion and binary decision diagram