# New algorithms and lower bounds for all-pairs max-flow in undirected graphs

## Abstract

We investigate the time-complexity of the All-Pairs Max-Flow problem: Given a graph with n nodes and m edges, compute for all pairs of nodes the maximum-flow value between them. If Max-Flow (the version with a given source-sink pair s,t) can be solved in time T(m), then an O(n2)·T(m) is a trivial upper bound. But can we do better? For directed graphs, recent results in fine-grained complexity suggest that this time bound is essentially optimal. In contrast, for undirected graphs with edge capacities, a seminal algorithm of Gomory and Hu (1961) runs in much faster time O(n) · T(m). Under the plausible assumption that Max-Flow can be solved in near-linear time m1+o(1), this half-century old algorithm yields an nm1+o(1) bound. Several other algorithms have been designed through the years, including Õ(mn) time for unit-capacity edges (unconditionally), but none of them break the O(mn) barrier. Meanwhile, no super-linear lower bound was shown for undirected graphs. We design the first hardness reductions for All-Pairs Max-Flow in undirected graphs, giving an essentially optimal lower bound for the node-capacities setting. For edge capacities, our efforts to prove similar lower bounds have failed, but we have discovered a surprising new algorithm that breaks the O(mn) barrier for graphs with unit-capacity edges! Assuming T(m) = m1+o(1), our algorithm runs in time m3/2+o(1) and outputs a cut-equivalent tree (similarly to the Gomory-Hu algorithm). Even with current Max-Flow algorithms we improve state-of-the-art as long as m = O(n5/3−ε). Finally, we explain the lack of lower bounds by proving a non-reducibility result. This result is based on a new quasi-linear time Õ(m) non-deterministic algorithm for constructing a cut-equivalent tree and may be of independent interest.