Mate is the opposite of Opponent. Opponent is the opposite of Ally. 
Minimize is related to Reduce. Reduce is related to Little. Little is a synonym of Slight. Slight is related to Shade. 
Mate <--Antonym--> Opponent <--Antonym--> Ally 
Minimize <--RelatedTo--> Reduce <--RelatedTo--> Little <--Synonym--> Slight <--RelatedTo--> Shade 