The Hirsch conjecture is true for (0,1)-polytopes
Mathematical Programming: Series A and B
Finding minimum-cost circulations by canceling negative cycles
Journal of the ACM (JACM)
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
New scaling algorithms for the assignment and minimum mean cycle problems
Mathematical Programming: Series A and B
A combinatorial interior point method for network flow problems
Mathematical Programming: Series A and B
A geometric Buchberger algorithm for integer programming
Mathematics of Operations Research
Handbook of combinatorics (vol. 2)
Variation of cost functions in integer programming
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A Variant of the Buchberger Algorithm for Integer Programming
SIAM Journal on Discrete Mathematics
A polynomial combinatorial algorithm for generalized minimum cost flow
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
An oracle-polynomial time augmentation algorithm for integer programming
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems
Journal of the ACM (JACM)
Relaxed Most Negative Cycle and Most Positive Cut Canceling Algorithms for Minimum Cost Flow
Mathematics of Operations Research
GRIN: An Implementation of Gröbner Bases for Integer Programming
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Test Sets and Inequalities for Integer Programs
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
Buchberger Algorithm and Integer Programming
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
0/1-Integer Programming: Optimization and Augmentation are Equivalent
ESA '95 Proceedings of the Third Annual European Symposium on Algorithms
Decomposition of Integer Programs and of Generating Sets
ESA '97 Proceedings of the 5th Annual European Symposium on Algorithms
Approximate local search in combinatorial optimization
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Discrete Optimization
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Primal methods constitute a common approach to solving (combinatorial) optimization problems. Starting from a given feasible solution, they successively produce new feasible solutions with increasingly 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 general framework for the design of polynomial-time algorithms for specific combinatorial optimization problems. We demonstrate its applicability by considering the min-cost flow problem, by giving a novel algorithm for linear programming over unimodular spaces, and by providing a different proof that for 0/1-integer programming an efficient algorithm solving the ordinary augmentation problem suffices for efficient optimization. Our main result also implies that directed augmentation is at least as hard as optimization. In other words, for an NP-hard problem already the task of finding a directed augmenting vector in polynomial time is hopeless, unless P = NP. We illustrate this kind of consequence with the help of the knapsack problem.