Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be the maximum number of edge-disjoint paths from $s$ to $t$ in $G$. Much research has gone into designing fast algorithms for computing connectivities in graphs. Previous work showed that it is possible to compute connectivities for all pairs of vertices in directed graphs with $m$ edges in $\tilde{O}(m^\omega)$ time [Chueng, Lau, and Leung, FOCS 2011], where $\omega \in [2,2.3716)$ is the exponent of matrix multiplication. For the related problem of computing "small connectivities," it was recently shown that for any positive integer $k$, we can compute $\min(k,\lambda(s,t))$ for all pairs of vertices $(s,t)$ in a directed graph with $n$ nodes in $\tilde{O}((kn)^\omega)$ time [Akmal and Jin, ICALP 2023]. In this paper, we present an alternate exposition of these $\tilde{O}(m^\omega)$ and $\tilde{O}((kn)^\omega)$ time algorithms, with simpler proofs of correctness. Earlier proofs were somewhat indirect, introducing an elegant but ad hoc "flow vector framework" for showing correctness of these algorithms. In contrast, we observe that these algorithms for computing exact and small connectivity values can be interpreted as testing whether certain generating functions enumerating families of edge-disjoint paths are nonzero. This new perspective yields more transparent proofs, and ties the approach for these problems more closely to the literature surrounding algebraic graph algorithms.

翻译：连通性（或等价地，无权最大流）是图论与组合优化中的一个重要度量。给定一个包含顶点 $s$ 与 $t$ 的图 $G$，从 $s$ 到 $t$ 的连通性 $\lambda(s,t)$ 定义为 $G$ 中从 $s$ 到 $t$ 的边不相交路径的最大数量。大量研究致力于设计计算图中连通性的快速算法。先前工作表明，可以在 $\tilde{O}(m^\omega)$ 时间内计算有向图中所有顶点对之间的连通性 [Chueng, Lau, and Leung, FOCS 2011]，其中 $\omega \in [2,2.3716)$ 是矩阵乘法的指数。对于计算“小连通性”这一相关问题，最近研究表明对于任意正整数 $k$，我们可以在 $\tilde{O}((kn)^\omega)$ 时间内计算具有 $n$ 个节点的有向图中所有顶点对 $(s,t)$ 的 $\min(k,\lambda(s,t))$ [Akmal and Jin, ICALP 2023]。本文中，我们对这些 $\tilde{O}(m^\omega)$ 与 $\tilde{O}((kn)^\omega)$ 时间算法提出了另一种阐述，并提供了更简洁的正确性证明。先前的证明在某种程度上是间接的，引入了一个优雅但特设的“流向量框架”来展示这些算法的正确性。相比之下，我们观察到这些计算精确连通性与小连通值的算法可被解释为测试某些枚举边不相交路径族的生成函数是否非零。这一新视角产生了更清晰的证明，并将处理这些问题的思路与代数图算法领域的文献更紧密地联系起来。