Distributed Algorithmic Mechanism Design and Algebraic Communication Complexity

  • Markus Bläser
  • Elias Vicari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)


In this paper, we introduce and develop the field of algebraic communication complexity, the theory dealing with the least number of messages to be exchanged between two players in order to compute the value of a polynomial or rational function depending on an input distributed between the two players. We define a general algebraic model, where the involved functions can be computed with the natural operations additions, multiplications and divisions and possibly with comparisons. We provide various lower bound techniques, mainly for fields of characteristic 0.

We then apply this general theory to problems from distributed mechanism design, in particular to the multicast cost sharing problem, and study the number of messages that need to be exchanged to compute the outcome of the mechanism. This addresses a question raised by Feigenbaum, Papadimitriou, and Shenker [9].


Rational Function Communication Complexity Equality Test Combinatorial Auction Transcendence Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abelson, H.: Towards a theory of local and global in computation. Theoret. Comput. Sci. 6(1), 41–67 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abelson, H.: Lower bounds on information transfer in distributed computations. J. Assoc. Comput. Mach. 27(2), 384–392 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosch, S.: Algebra, 3rd edn. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Briest, P., Krysta, P., Vöcking, B.: Approximation techniques for utilitarian mechanism design. In: Proc. ACM Symp. on Theory of Computing (2005)Google Scholar
  5. 5.
    Bürgisser, P., Clausen, M., Amin Shokrollahi, M.: Algebraic Complexity Theory. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bürgisser, P., Lickteig, T.: Test complexity of generic polynomials. J. Complexity 8, 203–215 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, P.: The communication complexity of computing differentiable functions in a multicomputer network. Theoret. Comput. Sci. 125(2), 373–383 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Hardness results for Multicast Cost Sharing. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 133–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Feigenbaum, J., Papadimitriou, C.H., Shenker, S.: Sharing the cost of a multicast transmission. J. Comput. Sys. Sci. 63, 21–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feigenbaum, J., Shenker, S.: Distributed algorithmic mechanism design: Recent results and future directions. In: Proc. 6th Int. Workshop on Discr. Alg. and Methods for Mobile Comput. and Communic., pp. 1–13 (2002)Google Scholar
  11. 11.
    Grigoriev, D.: Probabilistic communication complexity over the reals (preprint, 2007)Google Scholar
  12. 12.
    Hromkovic̆, J.: Communication Complexity and Parallel Computation. Springer, Heidelberg (1998)Google Scholar
  13. 13.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lehmann, D., O’Callaghan, L., Shoham, Y.: Truth revelation in approximately efficient combinatorial auctions. In: Proc. ACM Conference on Electronic Commerce (2003)Google Scholar
  15. 15.
    Luo, Z.-Q., Tsitsiklis, J.N.: Communication complexity of convex optimization. J. Complexity 3, 231–243 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Luo, Z.-Q., Tsitsiklis, J.N.: On the communication complexity of distributed algebraic computation. J. Assoc. Comput. Mach. 40(5), 1019–1047 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shafarevich, I.R.: Basic algebraic geometry 1 – Varieties in projective space, 2nd edn. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  18. 18.
    Yao, A.C.: Some complexity questions related to distributed computing. In: Proc. of 11th ACM Symp. on Theory of Comput., pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Markus Bläser
    • 1
  • Elias Vicari
    • 2
  1. 1.Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland

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