Subcubic algorithms for Gomory–Hu tree in unweighted graphs

ABSTRACTEvery undirected graph G has a (weighted) cut-equivalent tree T, commonly named after Gomory and Hu who discovered it in 1961. Both T and G have the same node set, and for every node pair s,t, the minimum (s,t)-cut in T is also an exact minimum (s,t)-cut in G. We give the first subcubic-time algorithm that constructs such a tree for a simple graph G (unweighted with no parallel edges). Its time complexity is Õ(n2.5), for n=|V(G)|; previously, only Õ(n3) was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using n−1 queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to Õ(√n) Max-Flow computations on n-node graphs.