Complexity and Algorithms for Euler Characteristic of Simplicial Complexes

TL;DR

This paper proves computing the Euler characteristic of simplicial complexes is #P-complete and introduces two new algorithms, based on combinatorial algebra and a non-algebraic approach, that outperform existing methods in speed.

Contribution

It establishes the computational complexity of the problem and provides two novel algorithms that significantly improve practical computation speed.

Findings

The problem is #P-complete.

Two new algorithms outperform previous implementations.

Algorithms are derived from combinatorial algebra and a non-algebraic approach.

Abstract

We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler characteristic. The two new algorithms are derived using combinatorial commutative algebra and we also give a second description of them that requires no algebra. We present experiments showing that the two new algorithms can be implemented to be faster than previous Euler characteristic implementations by a large margin.