The Complexity of Generic Primal Algorithms for Solving General Integer Programs

ngly better objective function value until an optimal solution is reached. From an abstract point of view, an augmentation problem is solved in each iteration. That is, given a feasible point, these methods find an augmenting vector, if one exists. Usually, augmenting vectors with certain properties are sought to guarantee the polynomial running time of the overall algorithm. In this paper, we show that one can solve every integer programming problem in polynomial time provided one can efficiently solve the directed augmentation problem. The directed augmentation problem arises from the ordinary augmentation problem by splitting each direction into its positive and its negative part and by considering linear objectives on these parts. Our main result states that in order to get a polynomial-time algorithm for optimization it is sufficient to efficiently find, for any linear objective function in the positive and negative part, an arbitrary augmenting vector. This result also provides a...

• Type: Journal
• Year: 2002
• Where: MOR
• Authors: Andreas S. Schulz, Robert Weismantel
• Keywords: Augmentation Problem, Directed Augmentation Problem, MOR 2002, Ordinary Augmentation Problem