Conditional Lower Bounds for All-Pairs Max-Flow

We provide evidence that computing the maximum flow value between every pair of nodes in a directed graph on $n$ nodes, $m$ edges,and capacities in the range $[1..n]$, which we call the All-Pairs Max-Flow problem, cannot be solved in time that is significantly faster (i.e., by a polynomial factor) than $O(n^3)$ even for sparse graphs. Since a single maximum $st$-flow can be solved in time $\tilde{O}(m\sqrt{n})$ [Lee and Sidford, FOCS 2014], we conclude that the all-pairs version might require time equivalent to $\tildeΩ(n^{3/2})$ computations of maximum $st$-flow,which strongly separates the directed case from the undirected one. Moreover, if maximum $st$-flow can be solved in time $\tilde{O}(m)$,then the runtime of $\tildeΩ(n^2)$ computations is needed. The latter settles a conjecture of Lacki, Nussbaum, Sankowski, and Wulf-Nilsen [FOCS 2012] negatively. Specifically, we show that in sparse graphs $G=(V,E,w)$, if one can compute the maximum $st$-flow from every $s$ in an input set of sources $S\subseteq V$ to every $t$ in an input set of sinks $T\subseteq V$ in time $O((|S| |T| m)^{1-ε})$,for some $|S|$, $|T|$, and a constant $ε>0$,then MAX-CNF-SAT with $n'$ variables and $m'$ clauses can be solved in time ${m'}^{O(1)}2^{(1-δ)n'}$ for a constant $δ(ε)>0$,a problem for which not even $2^{n'}/poly(n')$ algorithms are known. Such runtime for MAX-CNF-SAT would in particular refute the Strong Exponential Time Hypothesis (SETH). Hence, we improve the lower bound of Abboud, Vassilevska-Williams, and Yu [STOC 2015], who showed that for every fixed $ε>0$ and $|S|=|T|=O(\sqrt{n})$, if the above problem can be solved in time $O(n^{3/2-ε})$, then some incomparable conjecture is false. Furthermore, a larger lower bound than ours implies strictly super-linear time for maximum $st$-flow problem, which would be an amazing breakthrough.