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International Conference on Web and Internet Economics

WINE 2015: Web and Internet Economics pp 343–356Cite as

The VCG Mechanism for Bayesian Scheduling

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Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 9470))

Abstract

We study the problem of scheduling m tasks to n selfish, unrelated machines in order to minimize the makespan, where the execution times are independent random variables, identical across machines. We show that the VCG mechanism, which myopically allocates each task to its best machine, achieves an approximation ratio of \(O\left( \frac{\ln n}{\ln \ln n}\right) \). This improves significantly on the previously best known bound of \(O\left( \frac{m}{n}\right) \) for prior-independent mechanisms, given by Chawla et al. [STOC’13] under the additional assumption of Monotone Hazard Rate (MHR) distributions. Although we demonstrate that this is in general tight, if we do maintain the MHR assumption, then we get improved, (small) constant bounds for \(m\ge n\ln n\) i.i.d. tasks, while we also identify a sufficient condition on the distribution that yields a constant approximation ratio regardless of the number of tasks.

Supported by ERC Advanced Grant 321171 (ALGAME) and EPSRC grant EP/M008118/1. A full version of this paper can be found in [16].

Y. Giannakopoulos—A significant part of this work was carried out while the first author was a PhD student at the University of Oxford.

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Notes

  1. 1.

    Assume you run VCG on the first set of machines plus a dummy machine with processing time \(\beta \) on all tasks. The case where a task has processing time equal to \(\beta \) can be ignored without loss of generality for the case of continuous distributions.

  2. 2.

    We note here that for continuous distributions, such events of ties occurs with zero probability.

  3. 3.

    Here we use the following form, with \(\beta =1+\sqrt{5}\): for any \(\beta >0\), \(\mathrm {Pr}\left[ X\ge (1+\beta )\mu \right] \le e^{-\frac{\beta ^2}{2+\beta }\mu }\) for any binomial random variable with mean \(\mu \).

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Authors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Yiannis Giannakopoulos

  2. University of Oxford, Oxford, UK

    Maria Kyropoulou

Authors
  1. Yiannis Giannakopoulos
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  2. Maria Kyropoulou
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Correspondence to Maria Kyropoulou .

Editor information

Editors and Affiliations

  1. Athens University of Economics and Business, Athens, Greece

    Evangelos Markakis

  2. CWI and VU Amsterdam, Amsterdam, The Netherlands

    Guido Schäfer

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Giannakopoulos, Y., Kyropoulou, M. (2015). The VCG Mechanism for Bayesian Scheduling. In: Markakis, E., Schäfer, G. (eds) Web and Internet Economics. WINE 2015. Lecture Notes in Computer Science(), vol 9470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48995-6_25

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  • DOI: https://doi.org/10.1007/978-3-662-48995-6_25

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