Our work concerns algorithms for an unweighted variant of Maximum Flow. In the All-Pairs Connectivity (APC) problem, we are given a graph $G$ on $n$ vertices and $m$ edges, and are tasked with computing the maximum number of edge-disjoint paths from $s$ to $t$ (equivalently, the size of a minimum $(s,t)$-cut) in $G$, for all pairs of vertices $(s,t)$. Although over undirected graphs APC can be solved in essentially optimal $n^{2+o(1)}$ time, the true time complexity of APC over directed graphs remains open: this problem can be solved in $\tilde{O}(m^\omega)$ time, where $\omega \in [2, 2.373)$ is the exponent of matrix multiplication, but no matching conditional lower bound is known. We study a variant of APC called the $k$-Bounded All Pairs Connectivity ($k$-APC) problem. In this problem, we are given an integer $k$ and graph $G$, and are tasked with reporting the size of a minimum $(s,t)$-cut only for pairs $(s,t)$ of vertices with a minimum cut size less than $k$ (if the minimum $(s,t)$-cut has size at least $k$, we just report it is "large" instead of computing the exact value). We present an algorithm solving $k$-APC in directed graphs in $\tilde{O}((kn)^\omega)$ time. This runtime is $\tilde O(n^\omega)$ for all $k$ polylogarithmic in $n$, which is essentially optimal under popular conjectures from fine-grained complexity. Previously, this runtime was only known for $k\le 2$ [Georgiadis et al., ICALP 2017]. We also study a variant of $k$-APC, the $k$-Bounded All-Pairs Vertex Connectivity ($k$-APVC) problem, which considers internally vertex-disjoint paths instead of edge-disjoint paths. We present an algorithm solving $k$-APVC in directed graphs in $\tilde{O}(k^2n^\omega)$ time. Previous work solved an easier version of the $k$-APVC problem in $\tilde O((kn)^\omega)$ time [Abboud et al, ICALP 2019].

翻译：本文研究最大流无权重变体的算法。在全对连通性（APC）问题中，给定一个包含 $n$ 个顶点和 $m$ 条边的图 $G$，需要计算 $G$ 中从 $s$ 到 $t$ 的边不相交路径的最大数量（等价于最小 $(s,t)$-割的大小），针对所有顶点对 $(s,t)$。尽管无向图上的 APC 可以在本质最优的 $n^{2+o(1)}$ 时间内求解，但有向图上的 APC 的真实时间复杂度仍是开放问题：该问题可在 $\tilde{O}(m^\omega)$ 时间内求解，其中 $\omega \in [2, 2.373)$ 是矩阵乘法的指数，但尚无匹配的条件性下界。我们研究 APC 的一个变体，称为 $k$ 有界全对连通性（$k$-APC）问题。该问题中，给定整数 $k$ 和图 $G$，仅需报告最小割大小小于 $k$ 的顶点对 $(s,t)$ 的最小割值（若最小 $(s,t)$-割大小至少为 $k$，则仅报告为“大”而无需计算精确值）。我们提出一个算法，能在 $\tilde{O}((kn)^\omega)$ 时间内求解有向图上的 $k$-APC。对于所有 $k$ 为 $n$ 的多对数函数的情况，该运行时间为 $\tilde O(n^\omega)$，这在精细复杂度下的主流猜想下本质最优。此前，该运行时间仅对 $k\le 2$ 已知 [Georgiadis 等，ICALP 2017]。我们还研究了 $k$-APC 的另一个变体，即 $k$ 有界全对顶点连通性（$k$-APVC）问题，该问题考虑内部顶点不相交路径而非边不相交路径。我们提出一个算法，能在 $\tilde{O}(k^2n^\omega)$ 时间内求解有向图上的 $k$-APVC。此前的工作在 $\tilde O((kn)^\omega)$ 时间内解决了 $k$-APVC 问题的一个简化版本 [Abboud 等，ICALP 2019]。