Gomory–Hu tree


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 | V | − 1 maximum flow computations.

Definition

Let G = be an undirected graph with c being the capacity of the edge respectively.
Then T is said to be a Gomory–Hu tree of G if
where
  1. Se and Te are the two connected components of T∖ in the sense that form a s-t cut in G, and
  2. c is the capacity of the cut in G.

    Algorithm

Gomory–Hu Algorithm

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 ∈ ET, w = λpq 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
  1. green circles are vertices of T.
  2. red and blue circles are the vertices in G'.
  3. grey vertices are the chosen s and t.
  4. red and blue coloring represents the s-t cut.
  5. dashed edges are the s-t cut-set.
  6. A is the set of vertices circled in red and B is the set of vertices circled in blue.

    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. Experimental results comparing these algorithms are reported in Source code is available .
Cohen et al. reports results on two parallel implementations of Gusfield's algorithm using OpenMP and MPI, respectively. Source code of these implementations is available here: .

History

The Gomory–Hu tree was introduced by R. E. Gomory and T. C. Hu in 1961.

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.