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How to Generate Randomized Roundings with Dependencies and How to Derandomize Them

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Algorithm Engineering

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9220))

Abstract

We give a brief survey on how to generate randomized roundings that satisfy certain constraints with probability one and how to compute roundings of comparable quality deterministically (derandomized randomized roundings). The focus of this treatment of this broad topic is on how to actually compute these randomized and derandomized roundings and how the different algorithms with similar proven performance guarantees compare in experiments and the applications of computing low-discrepancy point sets, low-congestion routing, the max-coverage problem in hypergraphs, and broadcast scheduling. While mostly surveying results of the last 5 years, we also give a simple, unified proof for the correctness of the different dependent randomized rounding approaches.

Work done while both authors were affiliated with the Max Planck Institute for Informatics, Saarbrücken, Germany. Supported by the German Science Foundation (DFG) through grants DO 749/4-1, DO 749/4-2, and DO 749/4-3 in the priority programme SPP 1307 “Algorithm Engineering”.

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Notes

  1. 1.

    Note that some earlier solutions for special cases exist, e.g., for sums of variables adding up to one [47] or the hypergraph discrepancy problem [14, 15], which is the rounding problem with all variables being 1 / 2 and the rounding errors defined by a binary matrix.

  2. 2.

    Note that, in fact, \(E[y]=x\) and \(y \in \{\lfloor x \rfloor , \lceil x \rceil \}\) is equivalent to saying that y is a randomized rounding of x.

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Acknowledgements

The authors are grateful to the German Science Foundation for generously supporting this research through their priority programme Algorithm Engineering, both financially and by providing scientific infrastructure. We are thankful to our colleagues in the priority programme for many stimulation discussions. A particular thank goes to our collaborators and associated members of the project, namely Carola Doerr née Winzen (University of Kiel, then MPI Saarbrücken, now Université Pierre et Marie Curie—Paris 6), Tobias Friedrich (MPI Saarbrücken, now University of Jena), Michael Gnewuch (University of Kiel, now University of Kaiserslautern), Peter Kritzer (University of Linz), Marvin Künnemann (MPI Saarbrücken), Friedrich Pillichshammer (University of Linz), and Thomas Sauerwald (MPI Saarbrücken, now University of Cambridge).

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Authors and Affiliations

  1. Ecole Polytechnique de Paris, Palaiseau, France

    Benjamin Doerr

  2. Royal Holloway, University of London, London, UK

    Magnus Wahlström

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Correspondence to Magnus Wahlström .

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  1. Kiel University , Kiel, Germany

    Lasse Kliemann

  2. Karlsruhe Institute of Technology , Karlsruhe, Germany

    Peter Sanders

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Doerr, B., Wahlström, M. (2016). How to Generate Randomized Roundings with Dependencies and How to Derandomize Them. In: Kliemann, L., Sanders, P. (eds) Algorithm Engineering. Lecture Notes in Computer Science(), vol 9220. Springer, Cham. https://doi.org/10.1007/978-3-319-49487-6_5

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