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A hybrid branch-and-bound approach for exact rational mixed-integer programming

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Abstract

We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP  solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the Miplib and Mittelmann libraries and for a new collection of numerically difficult instances.

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Notes

  1. Project-and-shift fails to produce finite bounds on two instances, swath1 and swath2. The conditions necessary for its applicability within the tree (see Sect. 3.4) were not satisfied by these instances.

  2. Of course, even with a very carful implementation and extensive testing, a certain risk of an implementation error remains (also in the underlying exact LP  solver and the software package for rational arithmetic). So, the exact objective values reported here come with no warranty.

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Acknowledgments

The authors would like to thank Tobias Achterberg for helpful discussions on how to best incorporate the exact MIP  features into SCIP. We would also like to thank Daniel Espinoza for his assistance with QSopt_ex, which included adding new functionalities and writing an interface for use within SCIP.

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    William Cook

  2. Zuse Institute Berlin, Takustr. 7, 14195 , Berlin, Germany

    Thorsten Koch & Kati Wolter

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, USA

    Daniel E. Steffy

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  1. William Cook
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  2. Thorsten Koch
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  3. Daniel E. Steffy
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  4. Kati Wolter
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Corresponding author

Correspondence to Kati Wolter.

Additional information

This Research was supported by NSF Grant CMMI-0726370, ONR Grant N00014-12-1-0030, and the DFG Priority Program 1307 “Algorithm Engineering”.

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Cook, W., Koch, T., Steffy, D.E. et al. A hybrid branch-and-bound approach for exact rational mixed-integer programming. Math. Prog. Comp. 5, 305–344 (2013). https://doi.org/10.1007/s12532-013-0055-6

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