  Given the success of Douglas--Rachford splitting (DRS), it is natural to ask
whether DRS can be generalized. Are there other 2 operator resolvent-splittings
sharing the favorable properties of DRS? Can DRS be generalized to 3 operators?
This work presents the answers: no and no. In a certain sense, DRS is the
unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is
impossible without lifting, where lifting roughly corresponds to enlarging the
problem size. The impossibility result further raises a question. How much
lifting is necessary to generalize DRS to 3 operators? This work presents the
answer by providing a novel 3 operator resolvent-splitting with provably
minimal lifting that directly generalizes DRS.
