  We study the problem of fairly and efficiently allocating indivisible chores
among agents with additive disutility functions. We consider the widely-used
envy-based fairness properties of EF1 and EFX, in conjunction with the
efficiency property of fractional Pareto-optimality (fPO). Existence (and
computation) of an allocation that is simultaneously EF1/EFX and fPO are
challenging open problems, and we make progress on both of them. We show
existence of an allocation that is
  - EF1+fPO, when there are three agents,
  - EF1+fPO, when there are at most two disutility functions,
  - EFX+fPO, for three agents with bivalued disutilities. These results are
constructive, based on strongly polynomial-time algorithms. We also investigate
non-existence and show that an allocation that is EFX+fPO need not exist, even
for two agents.
