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（等价于无权重最大流）是图论与组合优化中的重要度量。对于给定图$G$及其顶点$s$和$t$，从$s$到$t$的连通度$\lambda(s,t)$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快速算法以计算图中的连通性。现有工作表明，对于包含$m$条边的有向图，可在$\tilde{O}(m^\omega)$时间内计算所有顶点对之间的连通性[Chueng, Lau, and Leung, FOCS 2011]，其中$\omega \in [2,2.3716)$为矩阵乘法指数。针对相关"小连通性"计算问题，近期研究证明：对于任意正整数$k$，可在$\tilde{O}((kn)^\omega)$时间内计算具有$n$个节点的有向图中所有顶点对$(s,t)$的$\min(k,\lambda(s,t))$值[Akmal and Jin, ICALP 2023]。本文提出了一种替代性的阐述方式，以更简洁的正确性证明重构上述$\tilde{O}(m^\omega)$与$\tilde{O}((kn)^\omega)$时间算法。早期证明略显迂回，引入了精巧但临时的"流向量框架"来验证算法正确性。相比之下，我们发现这些用于计算精确与小规模连通值的算法，可被解读为检测枚举边不相交路径族的特定生成函数是否非零。这一新视角不仅提供了更明晰的证明，更将此类问题的求解方法紧密关联至代数图算法的研究体系。