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Kolmogorov Complexity and Combinatorial Methods in Communication Complexity

  • Marc Kaplan
  • Sophie Laplante
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

Abstract

We introduce a method based on Kolmogorov complexity to prove lower bounds on communication complexity. The intuition behind our technique is close to information theoretic methods [1,2]. Our goal is to gain a better understanding of how information theoretic techniques differ from the family of techniques that follow from Linial and Shraibman’s work on factorization norms [3]. This family extends to quantum communication, which prevents them from being used to prove a gap with the randomized setting.

We use Kolmogorov complexity for three different things: first, to give a general lower bound in terms of Kolmogorov mutual information; second, to prove an alternative to Yao’s minmax principle based on Kolmogorov complexity; and finally, to identify worst case inputs.

We show that our method implies the rectangle and corruption bounds [4], known to be closely related to the subdistribution bound [2]. We apply our method to the hidden matching problem, a relation introduced to prove an exponential gap between quantum and classical communication [5]. We then show that our method generalizes the VC dimension [6] and shatter coefficient lower bounds [7]. Finally, we compare one-way communication and simultaneous communication in the case of distributional communication complexity and improve the previous known result [7].

Keywords

Communication Complexity Kolmogorov Complexity Auxiliary Input Universal Turing Machine Deterministic Protocol 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Marc Kaplan
    • 1
  • Sophie Laplante
    • 1
  1. 1.LRIUniversité Paris-Sud XIOrsay CEDEXFrance

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