(Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

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 these gaps by providing algorithms, conditional lower bounds, and non-reducibility results. Notably, we show that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out $O(n^{4-\varepsilon})$ time algorithms for some small $\varepsilon>0$, under a hypothesis called NSETH. To obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized Las Vegas $O(m^{2+o(1)})$ time combinatorial algorithm. This is our main technical result, improving over the recent $O(m^{11/5+o(1)})$ time Monte Carlo 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.