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A New Lower Bound for Deterministic Truthful Scheduling

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Abstract

We study the problem of truthfully scheduling m tasks to n selfish unrelated machines, under the objective of makespan minimization, as was introduced in the seminal work of Nisan and Ronen (in: The 31st Annual ACM symposium on Theory of Computing (STOC), 1999). Closing the current gap of [2.618, n] on the approximation ratio of deterministic truthful mechanisms is a notorious open problem in the field of algorithmic mechanism design. We provide the first such improvement in more than a decade, since the lower bounds of 2.414 (for \(n=3\)) and 2.618 (for \(n\rightarrow \infty\)) by Christodoulou et al. (in: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2007) and Koutsoupias and Vidali (in: Proceedings of Mathematical Foundations of Computer Science (MFCS), 2007), respectively. More specifically, we show that the currently best lower bound of 2.618 can be achieved even for just \(n=4\) machines; for \(n=5\) we already get the first improvement, namely 2.711; and allowing the number of machines to grow arbitrarily large we can get a lower bound of 2.755.

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Notes

  1. After the conference version of our paper [17], Dobzinski and Shaulker [13] and Christodoulou et al. [9] uploaded manuscripts improving this lower bound to 3 and \(\sqrt{n-1}+1\), respectively. The latter result is remarkably the first superconstant lower bound for this problem.

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Authors and Affiliations

  1. Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany

    Yiannis Giannakopoulos

  2. TU Munich, Munich, Germany

    Alexander Hammerl

  3. LASIGE, Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal

    Diogo Poças

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  1. Yiannis Giannakopoulos
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  2. Alexander Hammerl
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  3. Diogo Poças
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Correspondence to Diogo Poças.

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A significant part of this work was done while Y. Giannakopoulos and D. Poças were members of the Operations Research group at TU Munich, supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). D. Poças was also supported by FCT via LASIGE Research Unit, ref. UIDB/00408/2020. A preliminary version of this paper appeared in SAGT 2020 [17]

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Giannakopoulos, Y., Hammerl, A. & Poças, D. A New Lower Bound for Deterministic Truthful Scheduling. Algorithmica 83, 2895–2913 (2021). https://doi.org/10.1007/s00453-021-00847-2

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