(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 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.