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Alternatively, implementability can be characterized by an optimality condition related to an induced matching problem between types and alternatives. In an important class of environments where types satisfy a generalized \u201cone-dimensionality\u201d property, implementability is equivalent to a simple monotonicity condition where \u201chigher\u201d types receive \u201chigher\u201d alternatives. Using this finding, we show that for a single object auction, more allocation rules are implementable if either the type space is less dense or the extent of preference interdependence is less severe. For environments without transfers, nontrivial implementation is possible only when preferences exhibit sufficient independence. 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