On a conditional inequality in Kolmogorov complexity and its applications in communication complexity
Abstract
Romashchenko and Zimand~\cite{romzim:c:mutualinfo} have shown that if we partition the set of pairs $(x,y)$ of $n$bit strings into combinatorial rectangles, then $I(x:y) \geq I(x:y \mid t(x,y))  O(\log n)$, where $I$ denotes mutual information in the Kolmogorov complexity sense, and $t(x,y)$ is the rectangle containing $(x,y)$. We observe that this inequality can be extended to coverings with rectangles which may overlap. The new inequality essentially states that in case of a covering with combinatorial rectangles, $I(x:y) \geq I(x:y \mid t(x,y))  \log \rho  O(\log n)$, where $t(x,y)$ is any rectangle containing $(x,y)$ and $\rho$ is the thickness of the covering, which is the maximum number of rectangles that overlap. We discuss applications to communication complexity of protocols that are nondeterministic, or randomized, or ArthurMerlin, and also to the information complexity of interactive protocols.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1905.00164
 Bibcode:
 2019arXiv190500164R
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Information Theory
 EPrint:
 15 pages, 1 figure