Let's say that we want to solve a classic transportation problem without capacities using the stepping stone method. (Problem definition: A bipartite graph with supply nodes a1...m, demand nodes b1...bn, and costs wij for the transport of goods from ai to bj. Find a flow which satisfies the demand of all b nodes at minimal cost).
The first sentence in the chapter on the stepping stone method in the script is:
The restriction matrix T of the equations Tx=(a,b)T of the linear program has the following structure:
Then the text goes on to prove some lemmas about the subsets of independent columns in the restriction matrix. The matrix isn't mentioned after that, and it isn't used in the example. I couldn't find another source explaining it, and the textbook (Combinatorial optimization by Korte and Vygen) doesn't even mention the stepping stone method.
What I don't understand is: What do the rows and columns in this restriction matrix represent? Why are we interested in the amount of independent columns in it? And how is it used in the stepping stone method?