Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

A New Lower Bound for Deterministic Truthful Scheduling

  • Published:
Algorithmica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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.

References

  1. Ashlagi, I., Dobzinski, S., Lavi, R.: Optimal lower bounds for anonymous scheduling mechanisms. Math. Oper. Res. 37(2), 244–258 (2012). https://doi.org/10.1287/moor.1110.0534

    Article  MathSciNet  MATH  Google Scholar 

  2. Auletta, V., De Prisco, R., Penna, P., Persiano, G.: The power of verification for one-parameter agents. J. Comput. Syst. Sci. 75(3), 190–211 (2009). https://doi.org/10.1016/j.jcss.2008.10.001

    Article  MathSciNet  MATH  Google Scholar 

  3. Archer, A., Tardos, É.: Truthful mechanisms for one-parameter agents. In: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science (FOCS), pp. 482–491, (2001). https://doi.org/10.1109/sfcs.2001.959924

  4. Chen, X., Donglei, D., Zuluaga, L.F.: Copula-based randomized mechanisms for truthful scheduling on two unrelated machines. Theory Comput. Syst. 57(3), 753–781 (2015). https://doi.org/10.1007/s00224-014-9601-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Chawla, S., Hartline, J.D., Malec, D., Sivan B.: Prior-independent mechanisms for scheduling. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), pp. 51–60, (2013). https://doi.org/10.1145/2488608.2488616

  6. Christodoulou, G., Kovács, A.: A deterministic truthful PTAS for scheduling related machines. SIAM J. Comput. 42(4), 1572–1595 (2013). https://doi.org/10.1137/120866038

    Article  MathSciNet  MATH  Google Scholar 

  7. Christodoulou, G., Koutsoupias, E., Kovács, A.: Mechanism design for fractional scheduling on unrelated machines. ACM Trans. Algorithms 6(2), 1–18 (2010). https://doi.org/10.1145/1721837.1721854

    Article  MathSciNet  MATH  Google Scholar 

  8. Christodoulou, G., Koutsoupias, E., Kovács A.: On the Nisan-Ronen conjecture for submodular valuations. In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pp. 1086–1096, (2020). https://doi.org/10.1145/3357713.3384299

  9. Christodoulou, G., Koutsoupias, E., Kovács, A.: On the Nisan-Ronen conjecture. CoRR, abs/2011.14434, 2020. [arXiv:2011.14434v3]

  10. Christodoulou, G., Koutsoupias, E., Vidali, A.: A lower bound for scheduling mechanisms. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1163–1170, (2007). https://doi.org/10.5555/1283383.1283508

  11. Christodoulou, G., Koutsoupias, E., Vidali, A.: A lower bound for scheduling mechanisms. Algorithmica 55(4), 729–740 (2009). https://doi.org/10.1007/s00453-008-9165-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Dhangwatnotai, P., Dobzinski, S., Dughmi, S., Roughgarden, T.: Truthful approximation schemes for single-parameter agents. SIAM J. Comput. 40(3), 915–933 (2011). https://doi.org/10.1137/080744992

    Article  MathSciNet  MATH  Google Scholar 

  13. Dobzinski, S., Shaulker, A.: Improved lower bounds for truthful scheduling. abs/2007.04362, 2020. [arXiv:2007.04362v2]

  14. Daskalakis, C., Weinberg, S.M.: Bayesian truthful mechanisms for job scheduling from bi-criterion approximation algorithms. In: Proceedings of the 26th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1934–1952, (2015). https://doi.org/10.1137/1.9781611973730.130

  15. Epstein, L., Levin, A., Van Stee, R.: A unified approach to truthful scheduling on related machines. Math. Oper. Res. 41(1), 332–351 (2016). https://doi.org/10.1287/moor.2015.0730

    Article  MathSciNet  MATH  Google Scholar 

  16. Filos-Ratsikas, A., Giannakopoulos, Y., Lazos, P.: The Pareto frontier of inefficiency in mechanism design. In: Proceedings of the 15th Conference on Web and Internet Economics (WINE). pp. 186–199, (2019). https://doi.org/10.1007/978-3-030-35389-6_14

  17. Giannakopoulos, Y., Hammerl, A., Poças, D.: A new lower bound for deterministic truthful scheduling. In: Proceedings of the 13th symposium on algorithmic game theory (SAGT), (2020). https://doi.org/10.1007/978-3-030-57980-7_15

  18. Giannakopoulos, Y., Kyropoulou, M.: The VCG mechanism for Bayesian scheduling. ACM Trans. Econ. Comput. 5(4), 19:1-19:16 (2017). https://doi.org/10.1145/3105968

    Article  MathSciNet  MATH  Google Scholar 

  19. Giannakopoulos, Y., Koutsoupias, E., Kyropoulou, M.: The anarchy of scheduling without money. Theor. Comput. Sci 778, 19–32 (2019). https://doi.org/10.1016/j.tcs.2019.01.022

    Article  MathSciNet  MATH  Google Scholar 

  20. Hall, L.A.: Approximation algorithms for scheduling. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 1–45. PWS Publishing Company, Boston (1997)

    Google Scholar 

  21. Koutsoupias, E.: Scheduling without payments. Theory Comput. Syst. 54(3), 375–387 (2014). https://doi.org/10.1007/s00224-013-9473-0

    Article  MathSciNet  MATH  Google Scholar 

  22. Koutsoupias, E., Vidali, A.: A lower bound of \(1+ \phi\) for truthful scheduling mechanisms. In: Proceedings of Mathematical Foundations of Computer Science (MFCS), pp. 454–464, (2007). https://doi.org/10.1007/978-3-540-74456-6_41

  23. Koutsoupias, E., Vidali, A.: A lower bound of 1+\(\phi\) for truthful scheduling mechanisms. Algorithmica 66(1), 211–223 (2013). https://doi.org/10.1007/s00453-012-9634-6

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuryatnikova, O., Vera, J.C.: New bounds for truthful scheduling on two unrelated selfish machines. Theory Comput. Syst. 64(2), 199–226 (2019). https://doi.org/10.1007/s00224-019-09927-x

    Article  MathSciNet  MATH  Google Scholar 

  25. Lavi, R., Swamy, C.: Truthful mechanism design for multidimensional scheduling via cycle monotonicity. Games Econ. Behav. 67(1), 99–124 (2009). https://doi.org/10.1016/j.geb.2008.08.001

    Article  MathSciNet  MATH  Google Scholar 

  26. Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1), 259–271 (1990). https://doi.org/10.1007/bf01585745

    Article  MathSciNet  MATH  Google Scholar 

  27. Lu P.: On 2-player randomized mechanisms for scheduling. In: Proceedings of the 5th international workshop on internet and network economics (WINE), pp. 30–41, (2009). https://doi.org/10.1007/978-3-642-10841-9_5

  28. Lu, P., Yu, C.: An improved randomized truthful mechanism for scheduling unrelated machines. In: Proceedings of the 25th international symposium on theoretical aspects of computer science (STACS), pp. 527–538, (2008). https://doi.org/10.4230/LIPIcs.STACS.2008.1314

  29. Lu, P., Yu, C.: Randomized truthful mechanisms for scheduling unrelated machines. In: Proceedings of the 4th International Workshop on Internet and Network Economics (WINE), pp. 402–413, (2008). https://doi.org/10.1007/978-3-540-92185-1_46

  30. Mu'alem, A., Schapira, M.: Setting lower bounds on truthfulness. Games Econ. Behav. 110, 174–193 (2018). https://doi.org/10.1016/j.geb.2018.02.001

    Article  MathSciNet  MATH  Google Scholar 

  31. Nisan, N.: Introduction to mechanism design (for computer scientists). In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, chapter 9. Cambridge University Press, Cambridge (2007). https://doi.org/10.1017/cbo9780511800481.011

  32. Nisan, N., Ronen, A.: Algorithmic mechanism design (extended abstract). In: The 31st Annual ACM symposium on Theory of Computing (STOC), pp. 129–140, (1999)

  33. Nisan, N., Ronen, A.: Algorithmic mechanism design. Games Econ. Behav. 35(1/2), 166–196 (2001). https://doi.org/10.1006/game.1999.0790

    Article  MathSciNet  MATH  Google Scholar 

  34. Penna, P., Ventre, C.: Optimal collusion-resistant mechanisms with verification. Games Econ. Behav. 86, 491–509 (2014). https://doi.org/10.1016/j.geb.2012.09.002

    Article  MathSciNet  MATH  Google Scholar 

  35. Saks M., Yu, L.: Weak monotonicity suffices for truthfulness on convex domains. In: Proceedings of the 6th ACM Conference on Electronic Commerce (EC), pp. 286–293, (2005). https://doi.org/10.1145/1064009.1064040

  36. Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2003). https://doi.org/10.1007/978-3-662-04565-7

    Book  Google Scholar 

  37. Ventre, C.: Truthful optimization using mechanisms with verification. Theor. Comput. Sci. 518, 64–79 (2014). https://doi.org/10.1016/j.tcs.2013.07.034

    Article  MathSciNet  MATH  Google Scholar 

  38. Yu, C.: Truthful mechanisms for two-range-values variant of unrelated scheduling. Theor. Comput. Sci. 410(21–23), 2196–2206 (2009). https://doi.org/10.1016/j.tcs.2009.02.001

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

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

Authors
  1. Yiannis Giannakopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Hammerl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diogo Poças
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diogo Poças.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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]

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-021-00847-2

Search

Navigation