  We consider the problem of evaluating certain types of functional aggregation
queries on relational data subject to additive inequalities. Such aggregation
queries, with a smallish number of additive inequalities, arise
naturally/commonly in many applications, particularly in learning applications.
We give a relatively complete categorization of the computational complexity of
such problems. We first show that the problem is NP-hard, even in the case of
one additive inequality. Thus we turn to approximating the query. Our main
result is an efficient algorithm for approximating, with arbitrarily small
relative error, many natural aggregation queries with one additive inequality.
We give examples of natural queries that can be efficiently solved using this
algorithm. In contrast, we show that the situation with two additive
inequalities is quite different, by showing that it is NP-hard to evaluate
simple aggregation queries, with two additive inequalities, with any bounded
relative error.
