Analyzing and improving a classical Betti number estimation algorithm

Recently, a classical algorithm for estimating the normalized Betti number of an arbitrary simplicial complex was proposed. Motivated by a quantum algorithm with a similar Monte Carlo structure and improved sample complexity, we give a more in-depth analysis of the sample complexity of this classical algorithm. To this end, we present bounds for the variance of the estimators used in the classical algorithm and show that the variance depends on certain combinatorial properties of the underlying simplicial complex. This new analysis leads us to propose an improvement to the classical algorithm which makes the "easy cases easier'', in that it reduces the sample complexity for simplicial complexes where the variance is sufficiently small. We show the effectiveness and limitations of these classical algorithms by considering Erdős-Renyi random graph models to demonstrate the existence of "easy" and "hard" cases. Namely, we show that for certain models our improvement almost always leads to a reduced sample complexity, and also produce separate regimes where the sample complexity for both algorithms is exponential.