Looking for approximation class between NPO and Exp-APX
As pointed out by Markus Bläser in the comments section, objective values must be positive integers, so they cannot be 0. Hence my problem A is not formally an NPO problem. Personally I think it makes sense to consider feasible solutions with 0 objective value in that particular problem, but using standard approximation hardness notions requires redefining the problem so that 0-objective solutions are just "not feasible". In fact, under that new definition, problem A turns out to be NPO-complete.