  This paper mainly focuses on a resource leveling variant of a two-processor
scheduling problem. The latter problem is to schedule a set of dependent UET
jobs on two identical processors with minimum makespan. It is known to be
polynomial-time solvable.
  In the variant we consider, the resource constraint on processors is relaxed
and the objective is no longer to minimize makespan. Instead, a deadline is
imposed on the makespan and the objective is to minimize the total resource use
exceeding a threshold resource level of two. This resource leveling criterion
is known as the total overload cost. Sophisticated matching arguments allow us
to provide a polynomial algorithm computing the optimal solution as a function
of the makespan deadline. It extends a solving method from the literature for
the two-processor scheduling problem.
  Moreover, the complexity of related resource leveling problems sharing the
same objective is studied. These results lead to polynomial or
pseudo-polynomial algorithms or NP-hardness proofs, allowing for an interesting
comparison with classical machine scheduling problems.
