  The Holstein model on a square lattice at half-filling has a well-established
finite temperature phase transition to an insulating state with long range
charge density wave (CDW) order. Because this CDW formation suppresses pairing,
a superconducting (SC) phase emerges only with doping. In this work, we study
the effects of dilution of the local phonon degrees of freedom in the Holstein
model while keeping the system at half filling. We find not only that the CDW
remains present up to a dilution fraction $f \sim 0.15$, but also that long
range pairing is stabilized with increasing $f$, resulting in a {\it
supersolid} regime centered at $f \approx 0.10$, where long range diagonal and
off-diagonal correlations coexist. Further dilution results in a purely SC
phase, and ultimately in a normal metal. Our results provide a new route to the
supersolid phase via the introduction of impurities at fixed positions which
both increase quantum fluctuations and also are immune to the competing
tendency to phase separation often observed in the doped case.
