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 $O(N m^{\omega-1} + n m g)$ time, where $N$ denotes the total number of simplices in $K$, $m$ denotes the number of edges in $K$, $n$ denotes the number of vertices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $\omega$ denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $\tilde{O}(m^\omega)$ time, the second algorithm runs in $O(N m^{\omega-1})$ time and the third algorithm runs in $\tilde{O}(N^2 g + N m g{^2} + m g{^3})$ time which is nearly quadratic time when $g=O(1)$. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^\omega)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $\tilde{O}(m^\omega)$ time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).

翻译：我们研究寻找最小同调基的问题，即在给定单纯复形$K$中，寻找能生成$\mathbb{Z}_2$系数下一维同调类的最轻循环集合。该问题在过去几年中得到了广泛研究。对于一般复形，目前最好的确定性算法由Dey等人提出，其时间复杂度为$O(N m^{\omega-1} + n m g)$，其中$N$表示$K$中单形总数，$m$表示$K$的边数，$n$表示$K$的顶点数，$g$表示$K$的一维同调群秩，$\omega$表示矩阵乘法指数。本文提出三种概念简单的随机化算法，用于计算一般单纯复形$K$的最小同调基。第一种算法运行时间为$\tilde{O}(m^\omega)$，第二种算法运行时间为$O(N m^{\omega-1})$，第三种算法运行时间为$\tilde{O}(N^2 g + N m g{^2} + m g{^3})$，当$g=O(1)$时接近二次时间复杂度。我们还研究了在具有$n$个顶点和$m$条边的无向图$G$中寻找最小循环基的问题。该问题目前已知的最佳算法运行时间为$O(m^\omega)$。我们提出的算法虽然高层描述更简洁但代价稍高，其运行时间为$\tilde{O}(m^\omega)$。我们还提供了计算一般加权复形最小同调基的实用实现方案。该实现大体基于本文描述的算法思想，但在实用子程序的使用上有所不同。其中较耗时的步骤采用了并行实现，从而有望解决规模扩展问题。我们将计算结果与当前已知的最先进实现（ShortLoop）进行了对比。