  To integer programming problems, computational algebraic approaches using
Grobner bases or standard pairs via the discreteness of toric ideals have been
studied in recent years. Although these approaches have not given improved time
complexity bound compared with existing methods for solving integer programming
problems, these give algebraic analysis of their structures. In this paper, we
focus on the case that the coefficient matrix is unimodular, especially on the
primal and dual minimum cost flow problems, whose structure is rather
well-known, but new structures can be revealed by our approach. We study the
Grobner bases and standard pairs for unimodular programming, and give the
maximum number of dual feasible bases in terms of the volume of polytopes. And
for the minimum cost flow problems, we characterize reduced Grobner bases in
terms of graphs, and give bounds for the number of dual (resp. primal) feasible
bases of the primal (resp. dual) problems: for the primal problems the minimum
and the maximum are shown to be 1 and the Catalan number
$\frac{1}{d}\tbinom{2(d-1)}{d-1}$, while for the dual problems the lower bound
is shown to be $\Omega(2^{\lfloor d/6\rfloor})$. To analyze arithmetic degrees,
we use two approaches: one is the relation between reduced Gr{\"o}bner bases
and standard pairs, where the corresponding relation on the minimum cost flow
-- between a subset of circuits and dual feasible bases -- has not been so
clear, the other is the results in combinatorics related with toric ideals.
