# On a conditional inequality in Kolmogorov complexity and its applications in communication complexity

## TL;DR

This paper extends a Kolmogorov complexity inequality related to mutual information from partitions to coverings with overlapping rectangles, and explores its applications in various communication complexity models.

## Contribution

It generalizes a known inequality to overlapping coverings and applies this to analyze communication and information complexity in different protocol types.

## Key findings

- Extended the inequality to overlapping rectangle coverings.
- Derived bounds on mutual information involving covering thickness.
- Applied results to nondeterministic, randomized, and Arthur-Merlin communication protocols.

## Abstract

Romashchenko and Zimand~\cite{rom-zim: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 Arthur-Merlin, and also to the information complexity of interactive protocols.