  The star formation rate per unit area correlates well with the gas surface
density for different types of galaxies. However, this Kennicutt-Schmidt law
has not yet been examined for a large, homogeneously selected sample of SMGs,
which could provide useful SF implementation information for models of massive
galaxy formation and evolution. We aim at determining the K-S law parameters
for the first time for a well-selected, statistical sample of SMGs. We used
ALMA to conduct a 0.2" resolution, 870 $\mu$m imaging survey of 40 SMGs in
COSMOS, which were initially selected at 1.1 mm. We analysed a sample of 32/40
target SMGs, for which our new ALMA 870 $\mu$m data provide information about
the spatial extent of dust emission, and all of which have dust-obscured SFR
and dust-based gas mass estimates available from our previous study. We divided
our sample into equally large subsamples of main-sequence objects and
starbursts, and found their K-S relations to be of the form $\Sigma_{\rm SFR}
\propto \Sigma_{\rm gas}^{0.81\pm0.01}$ and $\Sigma_{\rm SFR} \propto
\Sigma_{\rm gas}^{0.84\pm0.39}$, respectively. The slightly sub-linear K-S
slopes we derived suggest that the SF efficiency is nearly constant across the
$\Sigma_{\rm gas}$ range probed. Under the assumption of a Galactic CO-to-H$_2$
conversion factor for the whole sample, the MS SMGs obey a constant global SFE
of about 21% per 100 Myr, while that of starburst SMGs is about 27%. The
corresponding gas depletion times are $\sim480$ Myr and 370 Myr. On average,
our SMGs have $\Sigma_{\rm gas}\gtrsim10^{3.9}$ M$_{\odot}$ pc$^{-2}$, which
suggests that they are Eddington-limited. This is consistent with the
theoretical expectation of a linear K-S relation for such systems. However,
size measurements of the CO-emitting regions of SMGs, and the $\alpha_{\rm CO}$
values of SMGs are needed to further constrain their $\Sigma_{\rm gas}$ values.
