2.5.18. Soft variant

Constraint C π‘Žπ‘™π‘ π‘œ is a soft variant of constraint C π‘’π‘›π‘‘π‘Ÿπ‘¦ . Note that, from an academic point of view, a soft constraint C π‘Žπ‘™π‘ π‘œ = is usually defined with a cost variable that quantifies how much the constraint C π‘’π‘›π‘‘π‘Ÿπ‘¦ = is violated. We exceptionally breaks this rule when it seems to make sense from an application point of view. For instance, within the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint, we reference the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πšŽπš‘πšŒπšŽπš™πš_0 since it can be seen as a kind of relaxation of the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint where we allow to use value 0 several times.

EXAMPLE: As an example, one of the possible soft variant of constraint C π‘’π‘›π‘‘π‘Ÿπ‘¦ = πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš (i.e.,Β the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint enforces all variables of a collection to take distinct values) is the constraint C π‘Žπ‘™π‘ π‘œ = 𝚜𝚘𝚏𝚝_πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πšŸπšŠπš›, where the cost is the minimum number of variables that need to be unassigned to satisfy the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint.