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Romashchenko and Zimand~\cite{rom-zim:c:mutualinfo} have shown that if wepartition the set of pairs $(x,y)$ of $n$-bit strings into combinatorialrectangles, 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 $t(x,y)$ isthe rectangle containing $(x,y)$. We observe that this inequality can beextended to coverings with rectangles which may overlap. The new inequalityessentially 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 $t(x,y)$ isany rectangle containing $(x,y)$ and $\rho$ is the thickness of the covering,which is the maximum number of rectangles that overlap. We discuss applicationsto communication complexity of protocols that are nondeterministic, orrandomized, or Arthur-Merlin, and also to the information complexity ofinteractive protocols.

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