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Using symmetry to optimize over the Sherali–Adams relaxation

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Abstract

In this paper we examine the impact of using the Sherali–Adams procedure on highly symmetric integer programming problems. Linear relaxations of the extended formulations generated by Sherali–Adams can be very large, containing \(O\left( \left( {\begin{array}{c}n\\ d\end{array}}\right) \right) \) many variables for the level-\(d\) closure. When large amounts of symmetry are present in the problem instance however, the symmetry can be used to generate a much smaller linear program that has an identical objective value. We demonstrate this by computing the bound associated with the level 1, 2, and 3 relaxations of several highly symmetric binary integer programming problems. We also present a class of constraints, called counting constraints, that further improves the bound, and in some cases provides a tight formulation. A major advantage of the Sherali–Adams formulation over the traditional formulation is that symmetry-breaking constraints can be more efficiently implemented. We show that using the Sherali–Adams formulation in conjunction with the symmetry breaking tool isomorphism pruning can lead to the pruning of more nodes early on in the branch-and-bound process.

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Acknowledgments

The author would like to thank Ricardo Fukasawa, Jeff Linderoth, Sven Leyffer, and the anonymous referees for their helpful comments about this work.

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  1. Department of Industrial and Systems Engineering, University of Tennessee, Knoxville, USA

    James Ostrowski

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  1. James Ostrowski
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Correspondence to James Ostrowski.

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Ostrowski, J. Using symmetry to optimize over the Sherali–Adams relaxation. Math. Prog. Comp. 6, 405–428 (2014). https://doi.org/10.1007/s12532-014-0072-0

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