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)


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].


Communication Complexity Kolmogorov Complexity Auxiliary Input Universal Turing Machine Deterministic Protocol 
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© 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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