  Motivated by Bland's linear-programming generalization of the renowned
Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm,
we discuss three closely-related natural augmentation rules for linear and
integer-linear optimization. In several nice situations, we show that
polynomially-many augmentation steps suffice to reach an optimum. In
particular, when using "discrete steepest-descent augmentations" (i.e.,
directions with the best ratio of cost improvement per unit 1-norm length), we
show that the number of augmentation steps is bounded by the number of elements
in the Graver basis of the problem matrix, giving the first ever strongly
polynomial-time algorithm for $N$-fold integer-linear optimization. Our results
also improve on what is known for such algorithms in the context of linear
optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the
number of steps in the simplex method) and are closely related to research on
the diameters of polytopes and the search for a strongly polynomial-time
simplex or augmentation algorithm.
