All pair Maximum Flow

Question

Given a directed weighted graph, how to find the Maximum Flow ( or Minimum Edge Cut ) between all pairs of vertices.
The naive approach is simply to call a Max Flow algorithm like Dinic's, whose complexity is O((V^2)*E), for each pair.
Hence for all pairs it is O((V^4)*E).

Is it possible to reduce the complexity to O((V^3)*E) or to O(V^3) by some optimizations?

Oreopithecus

PS: This is not home work. – Oreopithecus

Have you looked into Gomory–Hu tree ? – Nautical

@Nautical : This is exactly want I wanted. Thanks! And can you post a link where the Gusfield's algorithm is described with examples and pseudocode? – Oreopithecus

cs.princeton.edu/~kt/cut-tree the code can be found by visiting the Experiments link. – Nautical

@Nautical Could you make the comment into an answer? – Cull

Answer 1

Gomory-Hu Tree does not work with directed graphs, putting that aside, Gomory-Hu Tree will form a Graph maximum flow by applying minimum cuts.

The time complexity is:

O(|V|-1 * T(minimum-cut)) = O(|V|-1 * O(2|V|-2)) ~ O(|V|^2)

* using an optimal minimum-cut algorithm (Max-Flow Min-Cut Reduction)

This example illustrate how Gomory-Hu Tree is constructed from a given Graph

Coset

Gomory-Hu tree does not work for directed graphs(unless capacity (u,v)=(v,u) for all arcs).The cuts are not symmetric submodular functions, which is required for Gomory-Hu tree to work. – Proposal

@ChaoXu Thank you for the comment, I'll note it in my answer. – Coset

You have T(min-cut) as O(2|V|-2) in your text. What algorithm is that? E-K is O(|V| * |E|^2) ~ O(|V|^5) and Dinic is O(|V|^2 * |E|) ~ O(|V|^4), leaving your Gomory-Hu build at O(|V|^5), not O(|V|^2). – Castiglione

Answer 2

Gomory-Hu tree does not work for directed weighted graph.

It is an open problem whether there exist an algorithm to solve all pair maximum flow faster than running n^2 maximum flows on directed graphs.

Proposal