  We consider the Kob-Andersen model, a cooperative lattice gas with kinetic
constraints which has been widely analyzed in the physics literature in
connection with the study of the liquid/glass transition. We consider the model
in a finite box of linear size $L$ with sources at the boundary. Our result,
which holds in any dimension and significantly improves upon previous ones,
establishes for any positive vacancy density $q$ a purely diffusive scaling of
the relaxation time $T_{\rm rel}$ of the system. Furthermore, as $q\downarrow
0$ we prove upper and lower bounds on $L^{-2} T_{\rm rel} (q,L)$ which agree
with the physicists belief that the dominant equilibration mechanism is a
cooperative motion of rare large droplets of vacancies. The main tools combine
a recent set of ideas and techniques developed to establish universality
results for kinetically constrained spin models, with methods from bootstrap
percolation, oriented percolation and canonical flows for Markov chains.
