  We derive an effective Hamiltonian for the two-dimensional Hubbard-Holstein
model in the regimes of strong electron-electron and strong electron-phonon
interactions by using a nonperturbative approach. In the parameter region where
the system manifests the existence of a correlated singlet phase, the effective
Hamiltonian transforms to a $t_1-V_1-V_2-V_3$ Hamiltonian for hard-core-bosons
on a checkerboard lattice. We employ quantum Monte Carlo simulations, involving
stochastic-series-expansion technique, to obtain the ground state phase
diagram. At filling $1/8$, as the strength of off-site repulsion increases, the
system undergoes a first-order transition from a superfluid to a diagonal
striped solid with ordering wavevector $\vec{Q}=(\pi/4,3\pi/4)$ or
$(\pi/4,5\pi/4)$. Unlike the one-dimensional situation, our results in the
two-dimensional case reveal a supersolid phase (corresponding to the diagonal
striped solid) around filling $1/8$ and at large off-site repulsions.
Furthermore, for small off-site repulsions, we witness a valence bond solid at
one-fourth filling and tiny phase-separated regions at slightly higher
fillings.
