The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge this gap by providing algorithms, conditional lower bounds, and non-reducibility results. Our main result is that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out $n^{4-o(1)}$ time algorithms under a hypothesis called NSETH. In particular, to obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized $O(m^{2+o(1)})$ time combinatorial algorithm, improving on the recent $O(m^{11/5+o(1)})$ time algorithm [Huang et al., STOC 2023] and matching their $m^{2-o(1)}$ lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem. More generally, our main technical contribution is the insight that $st$-cuts can be verified quickly, and that in most settings, $st$-flows can be shipped succinctly (i.e., with respect to the flow support). This is a key idea in our non-reducibility results, and it may be of independent interest.

翻译：全对最大流问题在过去二十年中获得了广泛关注，其细粒度复杂度的许多结果已为人所知。尽管如此，该问题若干基本变体的时间复杂度仍存在巨大空白。本文旨在通过提供算法、条件下界与不可归约性结果来填补这一空白。我们的主要结论是：在大多数问题设定下，基于强指数时间假设(SETH)的确定性归约无法在NSETH假设下排除$n^{4-o(1)}$时间算法。特别地，针对单位节点容量无向图设定，我们设计了一种新的随机$O(m^{2+o(1)})$时间组合算法，改进了近期$O(m^{11/5+o(1)})$时间算法[Huang等人，STOC 2023]，并匹配其$m^{2-o(1)}$下界（至多亚多项式因子），从而基本确定了该问题设定下的时间复杂度。更广泛而言，我们的主要技术贡献在于发现$st$-割可快速验证，且在大多数设定下$st$-流可简洁传输（即关于流支撑）。这构成了我们不可归约性结果的核心思想，并可能具有独立研究价值。