Skip to main content
SpringerLink
Log in

Direction Preserving Zero Point Computing and Applications

(Extended Abstract)

  • Conference paper
Internet and Network Economics (WINE 2009)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 5929))

Included in the following conference series:

  • 2394 Accesses

Abstract

We study the connection between the direction preserving zero point and the discrete Brouwer fixed point in terms of their computational complexity. As a result, we derive a PPAD-completeness proof for finding a direction preserving zero point, and a matching oracle complexity bound for computing a discrete Brouwer’s fixed point.

Building upon the connection between the two types of combinatorial structures for Brouwer’s continuous fixed point theorem, we derive an immediate proof that TUCKER is PPAD-complete for all constant dimensions, extending the results of Pálvölgyi for 2D case [20] and Papadimitriou for 3D case [21]. In addition, we obtain a matching algorithmic bound for TUCKER in the oracle model.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arrow, K., Debreu, G.: Existence of an equilibrium for a competetive economy. Econometrica 22, 265–290 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, X., Deng, X.: On algorithms for discrete and approximate brouwer fixed points. In: STOC, pp. 323–330 (2005); Journal version appeared at: Chen, X., Deng, X.: Matching algorithmic bounds for finding a Brouwer fixed point. J. ACM 55(3) (2008)

    Google Scholar 

  3. Chen, X., Deng, X.: 3-NASH is PPAD-Complete. Electronic Colloquium on Computational Complexity (ECCC) (134) (2005)

    Google Scholar 

  4. Chen, X., Deng, X.: On the Complexity of 2D Discrete Fixed Point Problem. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 489–500. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Chen, X., Deng, X.: Settling the Complexity of Two-Player Nash Equilibrium. In: FOCS, pp. 261–272 (2006)

    Google Scholar 

  6. Chen, X., Deng, X., Teng, S.-H.: Computing Nash Equilibria: Approximation and Smoothed Complexity. In: FOCS, pp. 603–612 (2006)

    Google Scholar 

  7. Cohen, D.I.A.: On the Combinatorial Antipodal-Point Lemmas. Journal of Combinatorial Theory, Series B (27), 87–91 (1979)

    Google Scholar 

  8. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: STOC (2006)

    Google Scholar 

  9. Daskalakis, C., Papadimitriou, C.H.: Three-Player Games Are Hard. Electronic Colloquium on Computational Complexity (ECCC) (139) (2005)

    Google Scholar 

  10. Deng, X., Qi, Q., Saberi, A.: On the Complexity of Envy-Free Cake Cutting, http://arxiv.org/abs/0907.1334

  11. Freund, R.M.: Variable dimension complexes. Part I: Basic theory. Math. Oper. Res. 9(4), 479–497 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Freund, R.M., Todd, M.J.: A constructive proof of Tucker’s combinatorial lemma. Journal of Combinatorial Theory, Series A 30(3), 321–325 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Iimura, T.: A discrete fixed point theorem and its applications. Journal of Mathematical Economics 39, 725–742 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iimura, T., Murota, K., Tamura, A.: Discrete fixed point theorem reconsidered. METR 9 (2004)

    Google Scholar 

  15. Kuhn, K.W.: Some combinatorial lemmas in topology. IBM J. Res. Dev 4(5), 518–524 (1960)

    Article  MATH  Google Scholar 

  16. Longueville, M., Živaljević, R.T.: The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics. Journal of Combinatorial Theory, Series A (113), 839–850 (2006)

    Google Scholar 

  17. Matoušek, J.: Using the Borsuk-Ulam Theorem. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  18. Matoušek, J.: A combinatorial proof of Kneser’s conjecture. Combinatorica 24, 163–170 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Megiddo, N., Papadimitriou, C.H.: On Total Functions, Existence Theorems and Computational Complexity. Theor. Comput. Sci. 81(2), 317–324 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pálvölgyi, D.: 2D-TUCKER is PPAD-complete (Preprint in ECCC)

    Google Scholar 

  21. Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 498–532 (1994)

    Google Scholar 

  22. Prescott, T., Su, F.E.: A constructive proof of Ky Fan’s generalization of Tucker’s lemma. Journal of Combinatorial Theory, Series A (111), 257–265 (2005)

    Google Scholar 

  23. Scarf, H.E.: The approximation of fixed points of a continuous mapping. SIAM J. Applied Mathematics 15, 997–1007 (1967)

    Article  MathSciNet  Google Scholar 

  24. Simmons, F.W., Su, F.E.: Consensus-halving via theorems of Borsuk-Ulam and Tucker. Mathematical Social Science (45), 15–25 (2003)

    Google Scholar 

  25. Su, F.E.: Rental Harmony: Sperner’s Lemma in Fair Division. Amer. Math. Monthly (106), 930–942 (1999)

    Google Scholar 

  26. Tucker, A.W.: Some topological properties of disk and sphere. In: Proceedings of the First Canadian Mathematical Congress, pp. 285–309 (1945)

    Google Scholar 

  27. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Hong Kong SAR, P.R. China

    Xiaotie Deng & Jie Zhang

  2. Department of Management Science and Engineering, Stanford University, Stanford, California, USA

    Qi Qi

Authors
  1. Xiaotie Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jie Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information and Systems, Sapienza University of Rome, Via Ariosto 25, 00185, Roma, Italy

    Stefano Leonardi

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Deng, X., Qi, Q., Zhang, J. (2009). Direction Preserving Zero Point Computing and Applications. In: Leonardi, S. (eds) Internet and Network Economics. WINE 2009. Lecture Notes in Computer Science, vol 5929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10841-9_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10841-9_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10840-2

  • Online ISBN: 978-3-642-10841-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation