Re: [boost] [BGL] Stoer–Wagner min-cut algorithm

Subject: Re: [boost] [BGL] Stoer–Wagner min-cut algorithm
From: Daniel Trebbien (dtrebbien_at_[hidden])
Date: 2010-07-05 13:51:41

> Hi Daniel,
> I wonder if your min-cut algorithm can be used to find max flow, since
> min-cut and max-flow are dual? I am currently using two max-flow
> implementations in BOOST (i.e, Goldberg and Kolmogorov). What is the
> complexity compared to the two max-flow implementations in BOOST. I am
> looking for fastest max flow implementation available.
> Another related question: does BOOST offer parallel implementation of any
> max-flow algorithm?
> Thanks
> Dan 		 	   		

Hi Dan,

I should have been more precise. Given two vertices s and t where s is
called the source and t is called the sink, the maximum flow from s to
t is equal to the weight of a minimum s-t cut. Thus, a maximum flow
algorithm can compute a minimum s-t cut of a graph. The Stoer–Wagner
algorithm which I implemented calculates a min-cut of the graph, which
is a slightly different problem.

A cut of an undirected graph G = (V, E) is a partition of the vertices
into two, non-empty sets X and Y = V - X. The weight w(X, Y) of a cut
is the number of edges between X and Y if G is unweighted, or the sum
of the weights of edges between X and Y if G is weighted. Given two
vertices s and t, an s-t cut is a cut (X, Y) that separates s and t;
that is, either s is in X and t is in Y or t is in X and s is in Y. A
minimum s-t cut is an s-t cut of minimal weight.

Historically, one way of computing a min-cut of a graph was to
calculate the max-flow for every pair of vertices (s, t) and simply
pick a minimum s-t cut of minimal weight. This would be a min-cut.

I think that it is correct to say that if a min-cut is (X, Y), then
for all x in X and y in Y, (X, Y) is also a minimum x-y cut with the
max-flow from x to y and vice versa (because G is undirected) being
the min-cut weight w(X, Y). I think that it is also correct to say
that the least maximum flow between any two vertices of a graph is the
weight of a min-cut.

Intuitively speaking, because maximum flow is calculated between a
given source and sink, then a maximum flow algorithm will be better
than an min-cut algorithm, which has to consider all pairs of
vertices. So, if you are given two vertices s and t and you want to
know the maximum flow from s to t, you should use a maximum flow
algorithm. Also, a min-cut (X, Y) of G as returned by the Stoer–Wagner
algorithm might not be a minimum s-t cut;  both s and t could be in X
or Y.

As far as parallel implementations of maximum flow algorithms, I do
not know whether Parallel BGL offers parallel maximum flow algorithm
implementations.