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(n³) even for sparse graphs, namely m = O(n); thus for general m, it cannot be solved significantly faster than O(n²m). Since a single maximum st-flow can be solved in time Õ (mn) [Lee and Sidford, FOCS 2014], we conclude that the all-pairs version might require time equivalent to Ω( n³/²) 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 Õ (m), then the runtime of Ω( n²) computations is needed. This is in contrast to a conjecture of Lacki, Nussbaum, Sankowski, and Wulff-Nilsen [FOCS 2012] that All-Pairs Max-Flow in general graphs can be solved faster than the time of O(n²) computations of maximum st-flow. 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 ⊆ V to every t in an input set of sinks T ⊆ V in time O((|S||T |m)¹−^ε ), for some |S|, |T | and a constant ε > 0, then MAX-CNF-SAT (maximum satisfiability of conjunctive normal form formulas) 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 2ⁿ/poly(n) algorithms are known. Such running time 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(n), if the above problem can be solved in time O(n³/2−^ε ), then some incomparable (and intuitively weaker) 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. In addition, we show that All-Pairs Max-Flow in uncapacitated networks with every edge-density m = m(n) cannot be computed in time significantly faster than O(mn), even for acyclic networks. The gap to the fastest known algorithm by Cheung, Lau, and Leung [FOCS 2011] is a factor of O(m^ω−¹/n), and for acyclic networks it is O(n^ω−¹), where ω is the matrix multiplication exponent. Finally, we extend our lower bounds to the version that asks only for the maximum-flow values below a given threshold (over all source-sink pairs).