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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 O(n 2 ) • 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 m 1+o(1) , this half-century old algorithm yields an nm 1+o(1) bound. Several other algorithms have been designed through the years, including O(mn) time for unit-capacity edges (unconditionally), but none of them break the O(mn) barrier. Meanwhile, no super-linear lower bound is known 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) = m 1+o(1) , our algorithm runs in time m 3/2+o(1) and outputs a cut-equivalent tree

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

“…Moreover, such a tree can be computed using n − 1 max-flow calls. 1 Since their work, substantial effort has gone into obtaining better GHtree algorithms, and faster algorithms are now known for many restricted graph classes, including unweighted graphs [BHKP07, KL15,AKT21b], simple graphs [AKT21c, AKT21a, LPS21, Zha21, AKT22], planar graphs [BSW15], surface-embedded graphs [BENW16], bounded treewidth graphs [ACZ98, AKT20], etc. (see the survey [Pan16]).…”

“…Our techniques also improve the bounds known for the GHtree problem in unweighted graphs, and even for simple graphs. For unweighted graphs, the best previous results were Õ(mn) obtained by Bhalgat et al [BHKP07] and by Karger and Levine [KL15], and an incomparable result that reduces the GHtree problem to O( √ m) max-flow calls [AKT21b]. There has been recently much interest progress on simple graphs as well [AKT21c, AKT21a, LPS21, Zha21, AKT22], with the current best running time being (m + n 1.9 ) 1+o(1) .…”