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Semidefinite relaxations of ordering problems

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Abstract

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element \(u\) is before \(v\) in the ordering. In the second case, the profit depends on whether \(u\) is before \(v\) and \(r\) is before \(s\). The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.

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Notes

  1. For exact numbers of the speed differences see http://www.cpubenchmark.net/.

  2. These instances can be downloaded from http://flplib.uwaterloo.ca/.

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Acknowledgments

We thank Gerhard Reinelt and Marcus Oswald from Universität Heidelberg for many fruitful discussions and practical hints regarding the polyhedral approach to LOP. We also acknowledge constructive comments by two anonymous referees that led to the current version of the paper.

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Authors and Affiliations

  1. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, 9020,  Klagenfurt, Austria

    P. Hungerländer & F. Rendl

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  1. P. Hungerländer
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  2. F. Rendl
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Correspondence to F. Rendl.

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This paper is dedicated to Claude Lemarechal on the occasion of his 65th birthday.

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Hungerländer, P., Rendl, F. Semidefinite relaxations of ordering problems. Math. Program. 140, 77–97 (2013). https://doi.org/10.1007/s10107-012-0627-7

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