Universal Gröbner basis associated with the maximum flow problem

We give a formulation of the maximum flow problem as an integer programming problem in the standard form. We characterize elementary vectors of the kernel lattice of the matrix coefficient in our formulation in terms of the combinatorial property of the graph, such as circuits and s–t paths, and we determine the universal Gröbner basis of the toric ideal associated with the maximum flow problem. Next, we examine the kernel lattice of the reduced incidence matrix of the digraph. Under suitable assumptions for the digraph, we prove that it is generated by all incidence vectors of s–t directed paths, which yield the generator of the toric ideal associated with the reduced incidence matrix of the digraph.