On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond

TL;DR

This paper explores augmentation algorithms for linear and integer-linear programming, introducing new bounds and a strongly polynomial-time algorithm for N-fold integer-linear optimization, advancing the understanding of polynomial bounds and algorithm efficiency.

Contribution

It presents the first strongly polynomial-time algorithm for N-fold integer-linear optimization using discrete steepest-descent augmentations, and improves bounds on augmentation steps in linear optimization.

Findings

Polynomially-many augmentation steps suffice in certain cases.

Bound on steps is related to the Graver basis size.

Enhanced bounds generalize previous simplex method results.

Abstract

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.