Anti-monotone Constraints

Definition

A constraint C is anti-monotone if and only if for all itemsets S and S′:

$${\rm{if}}\,S\, \supseteq S'\,{\rm{and}}\,S\,{\rm{satisfies}}\,C,\,{\rm{then}}\,S'\,{\rm{satisfies}}\,C.$$

Key Points

Anti-monotone constraints [1,2] possess the following nice property. If an itemset S satisfies an anti-monotone constraint C, then all of its subsets also satisfy C (i.e., C is downward closed). Equivalently, any superset of an itemset violating an anti-monotone constraint C also violates C. By exploiting this property, anti-monotone constraints can be used for pruning in frequent itemset mining with constraints. As frequent itemset mining with constraints aims to find itemsets that are frequent and satisfy the constraints, if an itemset violates an anti-monotone constraint C, all its supersets (which would also violate C) can be pruned away and their frequencies do not need to be counted. Examples of anti-monotone constraints include min(S. Price) ≥ $20 (which expresses that the...