  We analyze the density and size dependence of the relaxation time $\tau$ for
kinetically constrained spin systems. These have been proposed as models for
strong or fragile glasses and for systems undergoing jamming transitions. For
the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any
density $\rho<1$ and for the Knight model below the critical density at which
the glass transition occurs, we show that the persistence and the spin-spin
time auto-correlation functions decay exponentially. This excludes the
stretched exponential relaxation which was derived by numerical simulations.
For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form
$\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we
rigorously prove the power law scalings recently derived in \cite{JMS} while in
$d\geq 3$ we obtain upper and lower bounds consistent with findings therein.
Our results are based on a novel multi-scale approach which allows to analyze
$\tau$ in presence of kinetic constraints and to connect time-scales and
dynamical heterogeneities. The techniques are flexible enough to allow a
variety of constraints and can also be applied to conservative stochastic
lattice gases in presence of kinetic constraints.
