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We study the problem of fairly and efficiently allocating indivisible choresamong agents with additive disutility functions. We consider the widely-usedenvy-based fairness properties of EF1 and EFX, in conjunction with theefficiency property of fractional Pareto-optimality (fPO). Existence (andcomputation) of an allocation that is simultaneously EF1/EFX and fPO arechallenging open problems, and we make progress on both of them. We showexistence 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 areconstructive, based on strongly polynomial-time algorithms. We also investigatenon-existence and show that an allocation that is EFX+fPO need not exist, evenfor two agents.

New Algorithms for the Fair and Efficient Allocation of Indivisible Chores