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We consider the school choice problem allowing schools to have minimum (in addition to the standard maximum) quotas. Standard properties such as strategyproofness, fairness, and nonwastefulness become incompatible with minimum quotas. Taking strategyproofness as necessary, we introduce new definitions of nonwastefulness and fairness that are compatible with minimum quotas. We provide new mechanisms that satisfy all minimum quotas and that are as nonwasteful and as fair as possible. We usedoi:10.1145/2841226 fatcat:hhbrfjeg5fhkzjn24wyo43br6u