  We studied the variability of the linear polarization and brightness of the
$\gamma$-ray binary LS I +61 303. High-precision BVR photopolarimetric
observations were carried out with the Dipol-2 polarimeter on the 2.2 m
remotely controlled UH88 telescope at Mauna Kea Observatory and the 60 cm
Tohoku telescope at Haleakala Observatory (Hawaii) over 140 nights in
2016--2019. We determined the position angle of the intrinsic polarization
$\theta \simeq 11^\circ$, which can either be associated with the projection of
the Be star's decretion disk axis on the plane of sky, or can differ from it by
$90^\circ$. Using the Lomb-Scargle method, we performed timing analyses and
period searches of our polarimetric and photometric data. We found
statistically significant periodic variability of the normalized Stokes
parameters $q$ and $u$ in all passbands. The most significant period of
variability, $P_\text{Pol} = 13.244 \pm 0.012$ d, is equal to one half of the
orbital period $P_\text{orb} = 26.496$ d. Using a model of Thomson scattering
by a cloud that orbits the Be star, we obtained constraints on the orbital
parameters, including a small eccentricity $e<0.2$ and periastron phase of
$\phi_\text{p}\approx 0.6$, which coincides with the peaks in the radio, X-ray,
and TeV emission. These constraints are independent of the assumption about the
orientation of the decretion disk plane on the sky. We also extensively discuss
the apparent inconsistency with the previous measurements of the orbital
parameters from radial velocities. By folding the photometry data acquired
during a three-year time span with the orbital period, we found a linear phase
shift of the moments of the brightness maximum, confirming the possible
existence of superorbital variability.
