  We report molecular gas observations of IRAS 20100-4156 and IRAS 03158+4227,
two local ultraluminous infrared galaxies (ULIRGs) hosting some of the fastest
and most massive molecular outflows known. Using ALMA and PdBI observations, we
spatially resolve the CO(1-0) emission from the outflowing molecular gas in
both and find maximum outflow velocities of $ v_{\rm max} \sim 1600$ and $\sim
1700$ km/s for IRAS 20100-4156 and IRAS 03158+4227, respectively. We find total
gas mass outflow rates of $\dot M_{\rm OF} \sim 670$ and $\sim 350$ Msun/yr,
respectively, corresponding to molecular gas depletion timescales $\tau^{\rm
dep}_{\rm OF} \sim 11$ and $\sim 16$ Myr. This is nearly 3 times shorter than
the depletion timescales implied by star formation, $\tau^{\rm dep}_{\rm SFR}
\sim 33$ and $\sim 46$ Myr, respectively. To determine the outflow driving
mechanism, we compare the starburst ($L_{*}$) and AGN ($L_{\rm AGN}$)
luminosities to the outflowing energy and momentum fluxes, using mid-infrared
spectral decomposition to discern $L_{\rm AGN}$. Comparison to other molecular
outflows in ULIRGs reveals that outflow properties correlate similarly with
$L_{*}$ and $L_{\rm IR}$ as with $L_{\rm AGN}$, indicating that AGN luminosity
alone may not be a good tracer of feedback strength and that a combination of
AGN and starburst activity may be driving the most powerful molecular outflows.
We also detect the OH 1.667 GHz maser line from both sources and demonstrate
its utility in detecting molecular outflows.
