  We study the phase diagram of the one-dimensional bosonic Hubbard model with
contact ($U$) and near neighbor ($V$) interactions focusing on the gapped
Haldane insulating (HI) phase which is characterized by an exotic nonlocal
order parameter. The parameter regime ($U$, $V$ and $\mu$) where this phase
exists and how it competes with other phases such as the supersolid (SS) phase,
is incompletely understood. We use the Stochastic Green Function quantum Monte
Carlo algorithm as well as the density matrix renormalization group to map out
the phase diagram. Our main conclusions are that the HI exists only at
$\rho=1$, the SS phase exists for a very wide range of parameters (including
commensurate fillings) and displays power law decay in the one body Green
function. In addition, we show that at fixed integer density, the system
exhibits phase separation in the $(U,V)$ plane.
