The Complexity of Extending Fair Allocations of Indivisible Goods

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. In view of the NP-hardness of the problem, we investigate whether - and under which conditions - one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that - unlike when computing allocations from scratch - the non-algorithmic question of whether more relaxed EF1 or EFX allocations exist can be completely resolved in the extension setting.