# The integral basis method for integer programming

@article{Haus2001TheIB, title={The integral basis method for integer programming}, author={Utz-Uwe Haus and Matthias K{\"o}ppe and Robert Weismantel}, journal={Mathematical Methods of Operations Research}, year={2001}, volume={53}, pages={353-361} }

Abstract. This paper introduces an exact algorithm for solving integer programs, neither using cutting planes nor enumeration techniques. It is a primal augmentation algorithm that relies on iteratively substituting one column by columns that correspond to irreducible solutions of certain linear diophantine inequalities. We demonstrate the algorithm's potential by testing it on some instances of the MIPLIB with up to 6000 variables.

## 36 Citations

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An exact primal augmentation algorithm for solving general linear integer programs that iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities is introduced.

### Polynomial Time Primal Integer Programming via Graver Bases

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We describe recent developments in primal methods for integer programming which lead to the polynomial time solution of new broad classes of integer programming problems with linear and various…

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A new algorithm for solving mixed integer programs that is an iterative technique for changing the representation of the original mixed integer optimization problem.

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A new primal algorithm for pure 0-1 problems based on strong valid inequalities is described and some encouraging computational results are given.

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One of the obstacles in a branch-and-cut IP-solving scheme is closing the gap between a known feasible solution and the solution(s) to LP relaxations of the problem. There are, however, a number of…

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The approach of integer pivoting is recalled and the family of Gomory-Young augmenta tion vectors that can be derived from a simplex tableau is introduced and a technique of co mbining Gomary-Young vectors and combinatorial augmentation vectors in one augm entation scheme is presented.

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Families of irreducible vectors of exponential size, derived from alternating cycles, where optimizing a linear function over each of these families can be done in polynomial time are studied.

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Examining progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the ``easy'' and ``hard'' sides of the complexity frontier concludes by citing briefly some of the more intriguing new avenues of research.

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