Fast Algorithms for Minimum Homology Basis

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N m^{ω-1} + n m g)$ time, where $N$ denotes the total number of simplices in $K$, $m$ denotes the number of edges in $K$, $n$ denote...