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. As a step towards ruling out even $mn^{1+\varepsilon-o(1)}$ SETH lower bounds for undirected graphs with unit node-capacities, we design a new randomized $O(m^{2+o(1)})$ time combinatorial algorithm. This is an improvement over 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：近二十年来，所有点对最大流问题（All-Pairs Max-Flow）备受关注，其细粒度复杂度研究已取得诸多成果。然而，对于该问题的若干基本变体的时间复杂度，仍存在显著认知空白。本文旨在通过提出算法、条件性下界及非可归约性结果来弥合这一差距。我们的主要结论是：在大多数问题设定中，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$n^{4-o(1)}$时间算法的存在。作为迈向排除无向单位节点容量图中$mn^{1+\varepsilon-o(1)}$量级SETH下界的一步，我们设计了一种新的随机化$O(m^{2+o(1)})$时间组合算法。该算法改进了近期$O(m^{11/5+o(1)})$时间算法[Huang等人，STOC 2023]的性能，并与其$m^{2-o(1)}$下界（至多差次多项式因子）相匹配，从而基本解决了该问题设定下的时间复杂度。更一般地，本文的核心技术贡献在于揭示了$st$-割可被快速验证，且在多数设定下$st$-流可被简洁描述（即基于流支撑集）。这一关键思想贯穿于我们的非可归约性结果，并可能具有独立研究价值。