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

Abstract

In 1961, Gomory and Hu showed that the max-flow values of all n 2 pairs of vertices in an undirected graph can be computed using only n − 1 calls to any max-flow algorithm, and succinctly represented them in a tree (called the Gomory-Hu tree later). Even assuming a lineartime max-flow algorithm, this yields a running time of O(mn) for this problem; with current max-flow algorithms, the running time is Õ(mn + n 5/2 ). We break this 60-year old barrier by giving an Õ(n 2 )-time algorithm for the Gomory-Hu tree problem, already with current max-flow algorithms. For unweighted graphs, our techniques show equivalence (up to poly-logarithmic factors in running time) between Gomory-Hu tree (i.e., all-pairs max-flow values) and a singlepair max-flow. * The first version of this paper (in November 2021) showed a slower time bound of Õ(n 23/8 ) for computing a Gomory-Hu tree. The only technical changes are in Section 2.