  In our companion paper \cite{Stojnicclupint19} we introduced a powerful
mechanism that we referred to as the Controlled Loosening-up (CLuP) for
handling MIMO ML-detection problems. It turned out that the algorithm has many
remarkable features and one of them, the \emph{computational complexity}, we
discuss in more details in this paper. As was explained in
\cite{Stojnicclupint19}, the CLuP is an iterative procedure where each
iteration amounts to solving a simple quadratic program. This clearly implies
that the key contributing factor to its overall computational complexity is the
number of iterations needed to achieve a required precision. As was also hinted
in \cite{Stojnicclupint19}, that number seems to be fairly low and in some of
the most interesting scenarios often not even larger than $10$. Here we provide
a Random Duality Theory based careful analysis that indeed indicates that a
very small number of iterations is sufficient to achieve an excellent
performance. A solid set of results obtained through numerical experiments is
presented as well and shown to be in a nice agreement with what the theoretical
analysis predicts. Also, as was the case in \cite{Stojnicclupint19}, we again
focus only on the core CLuP algorithm but do mention on several occasions that
the concepts that we introduce here are as remarkably general as those that we
introduced in \cite{Stojnicclupint19} and can be utilized in the analysis of a
large number of classes of algorithms applicable in the most diverse of
scientific fields. Many results in these directions we will present in several
of our companion papers.
