  In this paper we introduce the concept of additive approximation schemes and
apply it to load balancing problems. Additive approximation schemes aim to find
a solution with an absolute error in the objective of at most $\epsilon h$ for
some suitable parameter $h$. In the case that the parameter $h$ provides a
lower bound an additive approximation scheme implies a standard multiplicative
approximation scheme and can be much stronger when $h \ll$ OPT. On the other
hand, when no PTAS exists (or is unlikely to exist), additive approximation
schemes can provide a different notion for approximation.
  We consider the problem of assigning jobs to identical machines with lower
and upper bounds for the loads of the machines. This setting generalizes
problems like makespan minimization, the Santa Claus problem (on identical
machines), and the envy-minimizing Santa Claus problem. For the last problem,
in which the objective is to minimize the difference between the maximum and
minimum load, the optimal objective value may be zero and hence it is NP-hard
to obtain any multiplicative approximation guarantee. For this class of
problems we present additive approximation schemes for $h = p_{\max}$, the
maximum processing time of the jobs.
  Our technical contribution is two-fold. First, we introduce a new relaxation
based on integrally assigning slots to machines and fractionally assigning jobs
to the slots (the slot-MILP). We identify structural properties of
(near-)optimal solutions of the slot-MILP, which allow us to solve it
efficiently, assuming that there are $O(1)$ different lower and upper bounds on
the machine loads (which is the relevant setting for the three problems
mentioned above). The second technical contribution is a local-search based
algorithm which rounds a solution to the slot-MILP introducing an additive
error on the target load intervals of at most $\epsilon\cdot p_{\max}$.
