# Conditional Lower Bounds for All-Pairs Max-Flow

## TL;DR

This paper establishes strong computational lower bounds for the All-Pairs Max-Flow problem in directed graphs, indicating it likely cannot be solved significantly faster than cubic time, and connects this difficulty to the hardness of solving MAX-CNF-SAT, thus supporting the SETH conjecture.

## Contribution

It provides the first evidence that All-Pairs Max-Flow cannot be solved in strongly subcubic time, linking its complexity to the hardness of MAX-CNF-SAT and settling a conjecture about the problem's difficulty.

## Key findings

- All-Pairs Max-Flow likely requires near cubic time in directed graphs.
- A faster solution would imply a breakthrough in solving MAX-CNF-SAT.
- The results support the Strong Exponential Time Hypothesis (SETH).

## Abstract

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 $\tilde\Omega(n^{3/2})$ 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 $\tilde\Omega(n^2)$ 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 $O((|S| |T| m)^{1-\epsilon})$,for some $|S|$, $|T|$, and a constant $\epsilon>0$,then MAX-CNF-SAT with $n'$ variables and $m'$ clauses can be solved in time ${m'}^{O(1)}2^{(1-\delta)n'}$ for a constant $\delta(\epsilon)>0$,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 $\epsilon>0$ and $|S|=|T|=O(\sqrt{n})$, if the above problem can be solved in time $O(n^{3/2-\epsilon})$, 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.