We study the problem of finding a minimum homology basis, that is, a shortest 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 $O(N^\omega + N^2 g)$ time, where $N$ denotes the number of simplices 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 two 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, where $m$ denotes the number of edges in $K$, whereas the second algorithm runs in $O(m^\omega + N m^{\omega-1})$ time. 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.

中文翻译：

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最小循环基和最小同调基的快速算法

我们研究了寻找最小同调基的问题，即在给定的单纯复形 $K$ 中生成具有 $\mathbb{Z}_2$ 系数的 $1$ 维同源类的最短循环集。这个问题在过去几年中得到了广泛的研究。对于一般复形，目前最好的确定性算法，由 Dey 等人，在 $O(N^\omega + N^2 g)$ 时间内运行，其中 $N$ 表示 $K$, $g 中的单纯形数$表示$K$的$1$-同调群的秩，$\omega$表示矩阵乘法的指数。在本文中，我们提出了两个概念上简单的随机算法，它们计算一般单纯复形 $K$ 的最小同调基。第一个算法在 $\tilde{O}(m^\omega)$ 时间运行，其中 $m$ 表示 $K$ 中的边数，而第二个算法在 $O(m^\omega + N m^{\omega-1})$ 时间内运行。我们还研究了在具有 $n$ 个顶点和 $m$ 个边的无向图 $G$ 中找到最小循环基的问题。这个问题最著名的算法在 $O(m^\omega)$ 时间内运行。我们的算法具有更简单的高级描述，但成本稍高，运行时间为 $\tilde{O}(m^\omega)$。