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We consider a multi-agent resource allocation setting in which an agent'sutility may decrease or increase when an item is allocated. We take the groupenvy-freeness concept that is well-established in the literature and presentstronger and relaxed versions that are especially suitable for the allocationof indivisible items. Of particular interest is a concept called groupenvy-freeness up to one item (GEF1). We then present a clear taxonomy of thefairness concepts. We study which fairness concepts guarantee the existence ofa fair allocation under which preference domain. For two natural classes ofadditive utilities, we design polynomial-time algorithms to compute a GEF1allocation. We also prove that checking whether a given allocation satisfiesGEF1 is coNP-complete when there are either only goods, only chores or both.

Almost Group Envy-free Allocation of Indivisible Goods and Chores