In combinatorial optimization, the Gomory–Hu tree of an undirected graph with capacities is a weighted tree that represents the minimum s-t cuts for all s-t pairs in the graph. The Gomory–Hu tree can be constructed in maximum flow computations. It is named for Ralph E. Gomory and T. C. Hu.

Definition

Let be an undirected graph with c(u,v) being the capacity of the edge (u,v) respectively. Then T is said to be a Gomory–Hu tree of G, if for each where

Algorithm

Gomory–Hu Algorithm if such X exists. Otherwise, go to step 6. s, t ∈ X and find a minimum s-t cut (A′, B′) in G'. (u, v) . Output T.

Analysis

Using the submodular property of the capacity function c, one has Then it can be shown that the minimum s-t cut in G' is also a minimum s-t cut in G for any s, t ∈ X . To show that for all for some p ∈ P , q ∈ Q throughout the algorithm, one makes use of the following lemma, The lemma can be used again repeatedly to show that the output T satisfies the properties of a Gomory–Hu Tree.

Example

The following is a simulation of the Gomory–Hu's algorithm, where

Implementations: Sequential and Parallel

Gusfield's algorithm can be used to find a Gomory–Hu tree without any vertex contraction in the same running time-complexity, which simplifies the implementation of constructing a Gomory–Hu Tree. Andrew V. Goldberg and K. Tsioutsiouliklis implemented the Gomory-Hu algorithm and Gusfield algorithm, and performed an experimental evaluation and comparison. Cohen et al. report results on two parallel implementations of Gusfield's algorithm using OpenMP and MPI, respectively.

Related concepts

In planar graphs, the Gomory–Hu tree is dual to the minimum weight cycle basis, in the sense that the cuts of the Gomory–Hu tree are dual to a collection of cycles in the dual graph that form a minimum-weight cycle basis.